Elements of infinitesimal analysis

CHAPTER VIII

Elements of infinitesimal analysis In order to obtain the usual infinite probability spaces from finite spaces, and to work with infinite and infinitesimal numbers consistently, we shall use nonstandard analysis. This chapter is devoted to a brief introduction to its meth0ds.l I shall stress the intuitive ideas behind nonstandard (also called infinitesimal) analysis omitting many of the technical details and proofs. We assume no previous knowledge of nonstandard methods. We develop nonstandard analysis axiomatically, so that no deep understanding of mathematical logic is required.2 This chapter, however, contains a rather intuitive development, without attempting to be completely rigorous. The complete set of axioms, and a sketch of a proof that they are consistent, appears in Appendix A.3 Nonstandard analysis is a system for making precise statements that talk about a very large number of objects, and about numbers which are very close to zero. We repeat the example quoted in the Preliminaries. Let X be a random variable of mean p and finite variance u 2 . Consider v independent observations XI,Xz, . . . , X, of X. The law of large numbers can be fomulated intuitively by saying that if v is a large number, then almost surely for all large n 5 Y , the average (XI+. . . Xn)/nis approximately equal to p . In order to make this statement precise in usual mathematics, one replaces the finite sequence X I ,. . . , X, by an actually infinite sequence and constructs the infinite Cartesian product of the probability space s2 with itself, and, also, replaces ‘approximate equal’ by the appropriate E , S statement. The approach that uses nonstandard analysis is different. We retain, in this case, the finite

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‘For a more complete development see [71], [l],or [26]. My introduction is inspired mainly on 1711. ?Nonstandard analysis was first developed by A. Robinson in the sixties (see [78]).He used the method of enlargements and provided a construction of the nonstandard numbers. Nelson, [71],uses nonstandard set theory, [70],instead of enlargements. We shall follow Robinson and use the method of enlargements. The main mathematical techniques, however, especially in Chapter XI, are mainly in [71]. 3This axiomatic development appears, also, in [12]. 171

1 72

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

sequence X I , . . . , X, but we let v be an hyperfinite (infinite nonstandard) natural number, that is, a natural number that is larger than any finite natural number. We also replace ‘2 is approximately equal to y’ by ‘2- y is infinitesimal’, that is, ‘lz- yI is a positive number which is less than any real positive number’. Nonstandad analysis shows how to deal consistently with these objects: infinite natural numbers and infinitesimal numbers. In order t o achieve these aims, without constructing it explicitly here, we shall use an extension, *R of the real number system satisfying the following conditions z4 (1) Every mathematical notion which is meaningful for the system of real numbers is also meaningful for ‘R. In particular, ‘R is an ordered field extension of R. (2) Every mathematical statement formulated with the usual notions (which are these notions will be defined more precisely later) which is meaningful and true for the system of real numbers is meaningful and true also for *R: provided that we interpret any reference to sets or functions in the system ‘1not in terms of the totality of sets or functions, but in terms of a certain subset, called the family of internal sets or functions. (The properties of this families will be spelled out later.) For example, if the statement contains a phrase ‘for all sets of numbers’, we interpret this as ‘for all internal sets of numbers’. Similarly, the phrase ‘there exists a set of numbers’ as ‘there exists an internal set of numbers’. All elements of ‘R, however, are internal: the phrase ‘for all numbers’ is interpreted in *R as ‘for all elements of ‘R’. (3) The system of internal entities of ‘R has the following property: if S is an internal set, then all elements of S are also internal. (4) The set ‘R properly contains the set of real numbers, R. In particular, there is a positive element of ‘R which is smaller than all positive real numbers, i.e., an infinitesimal element. The subset of *R satisfying the property of being a natural number is denoted by *N. It is an internal set which properly contains N. In particular, there is an which, according to the order relation in ‘W,is greater than all individual in *N, numbers of W. The field ‘R is non-Archimedean, since it contains numbers that are greater than all numbers in R, which is a subfield of ‘R. Thus, for some number Q E ‘R 1 < a , 2 < a, . . . , n < a , ... ,

and so on

where n is any element of N. We have, however, that for any a E *R, there is a v E ‘M such that a < v. In fact, for any positive element a E *R, there is a v E *Nsuch that u 5a
‘For a construction of

‘B,see

Appendix A. Similar constructions appear in [78], [52] and

[I]. The conditions that follow are paraphrased from [78, pp. 50-561.

1. INFINITESIMALS

173

1. Infinitesimals

As explained above, we consider an extension of the set of natural numbers,

M,and hence of R, which has an infinite number u. That is, a natural number v that is larger than all elements of N.We denote the extended natural numbers by 'N,and the extended reals by *R, which we call the hyperreals. We also extend

all functions and relations defined on R to *R and assume that *R is an ordered field under the extended field operations and relations. Thus, l / v is smaller than any positive real number, and thus, it is an infinitesimal that is different from 0. We also assume that 'N is discretely ordered, that is, if n E *Nthen there is no element of *Nstrictly between n and n 1. We must notice that the addition of one infinite natural number, together with the requirement that *R be a field, means that many other infinite and infinitesimal numbers have to be added. We shall not construct *R hereI5but assume that we have done it and proceed axiomatically giving the main properties of this set that we shall need. A. Robinson, [78],was the first to prove that these additions can be done consistently. The following definitions introduce the basic concepts of infinitesimal analysis. These definitions make sense for any ordered field extending R. Notice that, since *R is an ordered field, the absolute value 1x1 of a number c makes sense and has the usual properties.

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DEFINITION VIII.l. Let x, y E '1.Then (1) t is finite if and only if 1x1 < n for some n E N. (2) t is infinite if and only if t is not finite, i.e., if and only if n < 1x1 for all n E N. x is infinite and positive is symbolyzed by t x 03, and x is infinite and negative by w -m. We write x << 03 to mean x $ 0 0 , and -m << t, to mean x -m. is infinitesimal if and only if1.1 < 1/n for every n E N. (3) (4) t is infinitely close to y, symbolized x x y,if and only if x - y is infinitesimal. ( 5 ) t 5 y if and only if c 5 y + E for some infinitesimal E . (6) x << y if and only if t < y and t $ y. (7) The elements of R are called standard and the elements of *R - R, non-

+

standard.

It is easy to show that w is an equivalence relation. It is also clear that the only infinitesimal that is a real number is 0. From these definitions, it follows immediately that 1.1 << 03 if and only if x is finite, and1.1 w 03 if and only if x is infinite. By assumption, there is an infinite element v in *R. It is almost immediate that 1/v is infinitesimal. Since *R is a field, 1/v E *R and, hence, *R contains infinitesimal elements. The following is a list of properties, which are deduced easily from the definitions.

PROPOSITION VIII. 1. 5For a sketch of the construction, see Appendix A .

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

174

( 1 ) x 0 if and only i f x is infinitesimaL (2) x w 0 if and only if for all positive reals r we have 1x1 < r . (3) Infinitesimals are finite. (4) Lei! x # 0. Then x B 0 if and only i f l/z i s infinite. ( 5 ) If x and y are finite, then so are x y and x y . (6) If x and y are infinitesimal, then so are x y and xy. (7) If x x 0 and Iyl< 00, then x y % 0. ( 8 ) x 5 y and y 5 x i f and only i f x w y . (9) For all n E ‘M, n is standard i f and only if n is finite. (10) The only standard infinitesimal is 0. (11) I f x and y are (standard) real numbers, then

+

x x y

e

+

x=y.

From these properties, we deduce:

PROPOSITION VI11.2. ( 1 ) x x y a n d z + O J -X x - Y. 2 % ( 2 ) x x y and 1.1 << 00 ux x uy. ( 3 ) x x z , y x z a n d t s u s y =j u x z . X P x 0, where p is a positive rational standard number. (4) x w 0 We also need relative approximate equality:

DEFINITION VIII.2. Let tx

y

E

# 0. Then we define ( e ) if and only if

X Y x -. E

E

Relative approximate equality has the same properties as approximate equality. In particular, all properties stated above are true when ‘=’ is replaced by ‘M

(E)’.

We shall occasionally use the standard part of a finite number. In any case, its introduction helps to understand the structure or the hyperreal line. We first prove:

PROPOSITION VIII.3 (STANDARD PART).Let a be afinite number. Then there is a unique real number r such that a w r . PROOF.Since the only real infinitesimal is 0, if r and s E R satisfy r x s, we have T = s. Hence, the unicity follows. We now show the existence. Let a be finite. Let r = sup{s E R 1 s 5 a } . This least upper bound exists, because a is finite. Suppose that r #ia . Then }r - a1 $ 0 and, hence, there is a real number s such that 0 < s 5 IT - a ( . If r >> a , then T - s 2 a, while if r << a, r s 5 a and, thus, in both cases, r is not a teast upper bound. Therefore, r x a.

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2. STANDARD, INTERNAL AND EXTERNAL SETS

175

In accordance with this theorem we can define the function st on the finite elements of + R to the real numbers by

st a = "a = the unique real r such that r w a.

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It is clear that for any finite number a, a = st a E , where E w 0. Thus, it is easy to show that st is a homomorphism from the finite elements of *R onto R. We can also introduce infinitesimal and infinite elements in other sets, besides the real numbers, such as "R or the complex numbers, @. As an example, I shall consider C. The set * @ can be considered as the set of all numbers x i y , where x,y E *R and i = A complex number z is infinitesimal, if 1.1 is infinitesimal as a real number. That is, the complex number z = x i y is infinitesimal, if and only if both x and y ape infinitesimal as real numbers. The complex numbers z and u are infinitely close, in symbols z % u, if and only if z - u is infinitesimal, i.e., Iz - UI M 0. Thus, if z1 = x1 i y l and 2 2 = z 2 iy2, then 11 % z2 if and only if X I w 2 2 and y1 w y2, i.e., the real and imaginary parts of zl and z2 are approximately equal. We can similarly extend the definition of infinite numbers.

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G.

+

+

+

We now introduce the definitions and notations for "'R. The elements of "'R are referred to as vectors and denoted Z. In general, we adopt the convention that 3c' = ( x 1 , x 2 , . . . ,x,), y'= ( y 1 , y 2 , .. . , y , ) , etc. The number n will usually be clear from the context. We denote by 1141the length of i,i.e.

where X I , . . . , xn are the components of Z. Thus, the distance between Z and y' can be written as 115- 41. We say that z' is real or standard, if all its components X I , . . . , x,, are real. We say that z' is infinitesimal, if 1141 is infinitesimal. We also say that 31' is approximately equal to f, in symbols, 3c' M y', if 112- 41 is infinitesimal. We have the following easy proposition:

PROPOSITION VIII.4. (1) I is infinitesimal if and only if its components,

XI,

finitesirnals. (2) I = y ' i f a n d o n l y i f x I x y l , x 2 ~ y 2 ., . . , I , x ~ , .

x2, . . . , x, are in-

2. Standard, internal and external sets

In order to incorporate all of standard mathematics, and prove our main theorems, we must distinguish between standard, internal, and external sets. We shall only introduce here the minimum required for our development; in Appendix A, a complete set of axioms is given, so that all of analysis can be obtained on its basis. We call the usual mathematical notions, not including those that are obtained from the notions of 'infinitesimal', 'standard', or any of the new notions introduced before, standard notions. In particular, the membership relation, E, is

176

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

standard. In fact, as it is done in Appendix A, we can just have the membership and equality relations as the only primitive standard notions, and the notion of ‘being standard’, as the only primitive nonstandard notion. All nonstandard notions can be defined with the latter notion. For instance, we can define ‘ 2 is infinitesmal’ by ‘ x < 1/n for every standard natural number n’. We need for our development the notions of standard and internal sets and functions. The standard sets we need are subsets of “*R, and the standard funtions which are needed are functions from subsets of “*R into *R. We also consider standard and internal sets which are families of these subsets or functions defined on these families into *R.Instead of defining standard objects, we shall give some of their properties. For a more complete study see Appendix A. In order to codify the sets we need, we introduce the following definition: For any set X, let Vt(X) =

x

Vn+l(X) = K(X) Up(Vn(X)). All objects that we need are in one of the sets Vm(*R), for a certain m. By the usual set-theoretic construction, the functions of interest are also in these sets. As an example we shall determine the level m where a function f : *R -+ *R is. Such a function is a set of ordered pairs of elements of *R, so we have to see where ordered pairs are. Let Q, b E ‘R. We have, using the customary definition of ordered pair (a,b) = { { Q } { ~ , b H .

The sets {a} and { a , b } are subsets of *R, and, hence, belong to Vl(*R). Thus, the ordered pair {a,b) is a subset of Vl(*E), and belongs to V,(*R). Therefore, f is a subset of V2(*R) and belongs to b(’R). Similarly, n-tuples of elements of ‘1,for n E N,which are functions from (1,. . . ,n) to *R, belong to V3(*R), and, thus, “*R, the set of all n-tuples of elements of ‘W, is in V4(*E). As another example, suppose that R E Vm(*R). A probability measure is a function Pr from subsets of R into ‘R, and, hence, Pr E Vm+4(*R). In a similar way, we can see that all sets, families of sets, or functions, can always be assumed to belong to one of these sets Vm(*R). We shall need the following properties of standard sets: ( I ) Real vectors, i.e., elements of “R (but not “R itself) are standard. (2) The sets *R and ‘M are standard. (3) If a set A can be defined as the subset of “*R or Vm(*R), for a certain m, that satisfies a condition which only involves standard notions, and other, already defined, standard objects (i.e., real numbers, standard sets or functions), then A or f are also standard. Thus, in order to define standard objects we do not use “infinitesimal”, “real”, “finite” (in its new sense), “infinite”, “R?, etc. (4) A finite set of standard objects is standard. Unions and intersections of a finite number of standard sets are standard. Sums, differences, products, quotients, compositions, and inverses of standard functions are standard.

2. STANDARD, INTERNAL AND EXTERNAL SETS

177

( 5 ) Iff is standard, then the domain off is standard, and, for every standard z in its domain, f(z) is also standard. In particular, if 3: E 1is in the domain of f , f(z) E R. (6) This is the main property of standard sets. Intuitively it says that a standard set is determined by its standard objects. For instance, if A and B are standard subsets of 'R that contain the same real numbers (i.e., A n R = B n R), then A = B. A similar property is valid for "R. In general, if A and B are standard sets that have the same standard objects, then A = B . (7) I f f and g are standard functions with the same domain and such that for each real 1: (or each standard z) in its domain, f(z) = g(z), then f = g. (8) Every subset of "R and every function on "W has a standard extension. In general, every set S of standard objects has a standard extension, 'S, such that the only standard objects in 'S are the elements of S. A similar statement is true for functions. Intervals with real endpoints are examples of standard sets. For instance, if a , b are standard real numbers, then

is standard. Elementary functions, are also standard. For instance,

is a standard function defined over all of 'R. It is clear by (6) that the standard extension of a set of standard objects is unique. It is easy to show that if a standard set A contains all the reals, then A = *R. Also, if A is a standard set of positive hyperreals which contains all positive reals, then A = *R+, the set of positive hyperreds. In general, we have the following proposition:

PROPOSITION VIII.5. If A and B are standard with A C B and A contains all the standard objects in B , then A = B . PROOF.Let A and B be standard, A C B , where A contains all the standard objects in B . Then, if z E B is standard, then c E A . Thus, all standard objects in B are in A . But A E B , and, hence, all standard objects in A are in B . Therefore, A and B contain the same standard objects. Since A and B are standard, A = B . 0 PROPOSITION VIII.6. If g and h are standard functions with the same domain A C "*R into m*R, f o r certain n and rn, and for every i? E A , g(Z) M h(i?), then g = h.

178

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

PROOF. If 2’ is real in A , then g ( 2 ) and h(2’) are real, and, as g(Z) w h(Z), we have g(2’) = h ( q . Thus, the standard set (5E A l g ( q = h ( q } contains all the real vectors in A . Since A is standard A = { Z E A l g ( Z ) = h(2)}. 0

The notion of internal set or function is wider than the corresponding standard The family of internal objects has the following properties. Every standard set is internal. Every element of an internal set is internal. A set or function is internal if it can be defined using only standard mathematical notions and other, already defined, internal objects (i.e., elements of n*R,internal sets or functions). T h a t is, just as for standard objects, in order to define internal objects we do not use “infinitesimal”, etc. On the other hand, we may use for “real”, “finite”, “infinite”, “d’, defining an internal set, elements of *R and not just of R. The union and intersection of an internal family of internal sets is internal and the family of the internal subsets of an internal set, is internal. Sums, differences, products, quotients, composition and inverse of internal functions are internal. A set or function which is not internal is called external. We can conclude from (1) and (2), that all elements of n*R, and ”‘R itself, are internal. The intervals of *R are instances of internal sets. For example, if a, b E *R, not necessarily real (for instance a and b may be infinitesimals) the interval [a,b] can be defined by [a, b] = (2 E *R I a 5 2 5 b } . The condition which defines [a,b], a 5 2 5 b, only has usual mathematical notions, in this definition a and b occur, which are elements of *R, and the only set that occurs, in this case ‘R, is internal. On the other hand, as we shall show later, the set of all infinitesimals is not internal. We can see that in its definition we use the notion of “infinitesimal”, so that the fact that it is not internal does not violate the rules given above. All elementary functions are standard, and hence, internal, because they can be defined with standard notions and particular real numbers. For instance, the function 1 if 2 # 0 f(z)= ,; is standard because it can be defined with standard notions and the number zero. Another internal function, but this time not standard, is f ( 2 ) = E for all 2: E ‘24, where E 0. On the other hand, the function st defined in the previous section is external. It needs for its definition the notions of “real” (or “standard”) and c‘d’. \Ve assume the following principle, called the transfer principle:

=

2. STANDARD, INTERNAL AND EXTERNAL SETS

179

Every standard theorem is true for standard objects. In Appendix A, we indicate how to prove this assumption from our axioms. As was stated in the introduction to this chapter, every phrase of the form ‘there is a set’ or ‘for every set’ should be replaced, in the statement of each standard theorem, by ‘there is an internal set’ or ‘for every internal set’. Similar changes have to be done for functions. Most of the standard theorems we shall use do not contain these types of phrases, so that we don’t need to worry about them. The least upper bound axiom is not true for external subsets of *R. For instance, 1is a bounded (in *R)subset of *R which does not have a least upper bound. On the other hand, although we shall not use it, the least upper bound principle is valid for internal subsets. Thus, R is external. We also need some properties of *N.

(1) N c *N. (2) The finite elements of *Nare exactly the elements of M. (3) *Nis a proper extension of M. That is, there are elements in *Nwhich are not in N. These elements of *N- M are infinite. (4) For every positive hyperreal number z E *R, there is a natural number v E *N, such that v - 1 < 1: 5 v.

(5) *M satisfies the Internal Induction Principle, that is: Internal Induction Principle: If S is an internal subset of *N that satisfies the following conditions: (a) 1 E S; (b) n € S n+l~S; then we have S = *M. Note that S must be internal.

*

With these properties, it is easy to show that internal subsets of *Nsatisfy all standard conditions satisfied by subsets of N in usual mathematics. For instance, every nonempty internal subset of *Nhas a least element, and every nonempty internal bounded subset of *M has a largest element. The induction principle, without the restriction on internal sets, is, of course, valid for N. We shall call this induction principle for M, the etternal induction principle. It is also clear that l€N and

n€N

n + l ~ N .

Thus, if N were internal, by the internal induction principle, N would be equal to *M. But we know it is not. Therefore, W is external. We thus deduce the overflour principles, “781.

THEOREM VIII.7 (OVERFLOW PRINCIPLES).

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

If an internal set A contains all finite natural numbers larger than a

certain finite natural number n, then A also contains all infinite natural numbers less than a certain infinite natural number v. If an internal set A contains all infinite natural numbers less than a certain v M 00, then it contains all finite natuml number larger than a certain finite number n. If an internal set A contains all positive infinitesimals, then A contains a positive real number d . If an internal set A contains all positive noninfinitesimal numbers less than a certain real number, then A contains all infinitesimals larger than a certain infinitesimal. (Standard overflow). If a standard subset A of *W i s nonempty, then it contains a finite (standard) real number. The third overflow principle shows that the set of infinitesimals is external.

PROOF OF (1). Suppose that A is internal and let A contains all finite natural numbers larger than the finite natural number n, and B = A U { m E *NI m 5 n}. Then B is internal. If *N- B is empty, then the theorem is obviously true. So, in order to complete the proof, suppose that it is nonempty. The set *N- B is internal, and, hence by the internal induction principle, since it is assumed t o be nonempty, it has a minimal member. Let v 1 be that minimal element, which cannot be finite. Then v satisfies the conditions of the theorem,

+

PROOF OF (2). Suppose that A satisfies the hypothesis of (2). Then the set B = A U {n E *NI n 2 v} is internal. If *N-B is empty, we are done. Let, then, *N- B # 8. Then *N- B is an internal set of finite numbers, because all infinite numbers are in B . Thus, *N- B is bounded and, hence, it contains a maximal element, n. The number n , which must be finite, is the required number. 0 PROOF OF (3). Let B be the internal subset of *Ndefined by B = {v I(0, L) C A } . v From the hypothesis of (3) we deduce that B contain all infinite natural numbers. Then, by (2), B contains a finite natural number n. Take d = l / n . (4)is obtained from (I), as (3) is from (2). 0

PROOFOF (5). Suppose that A is a standard subset of *R which does not contain real numbers. The empty set, 0, is obviously standard and contains the same standard reals as A , namely none. Hence, A = 8. As a simple example of the use of overflow, we shall prove the following propo-

si tions.

PROPOSITION VIIl.8. A number x = 0 if and only if 1x1 nonstandard (infinite) natural number v.

< 1/v

for some

2. STANDARD, INTERNAL AND EXTERNAL SETS

181

PROOF.It is clear that if 1x1 < 1/v for a certain infinite v, then 1x1 < l / n , for every finite natural number n, because a finite number is always less than an infinite number. So, in order to prove the converse, assume that 1x1 natural number n. Then, the set

<

1/n for every finite

1 A = {n E *NI 1x1 < -}

n is internal and contain all finite numbers. Hence, A contains an infinite number, v. This is the required number. [7

PROPOSITION VIII.9. Let f : S --+ *R be an internal function such that for every x E S, f(x) is finite. Then there is a finite number m such that f(x) 5 m, for eve y x E S. The proposition asserts that an internal function that has only finite values is bounded by a finite number.

PROOF. The set {n E *NI f(x) 5 n, for every x E S} is internal and contains all infinite natural numbers. Hence, by overflow, it contains a finite natural number rn. This is a bound of f. 0 We also need a lemma proved by A. Robinson, [78].

LEMMAVIII.10 (ROBINSON’S LEMMA). Let f be an internal function such that f o r every finite x 2 Q in its domain, where a is finite, f(x) M 0. Then there is an infinite b such that, for all x with Q 5 x 5 b, we have f (x) M 0. In particular, we have: Let X I , 2 2 , . . . , x,,, . . ., x,,, f o r n 5 ,u M oa,be an internal sequence. If xi M 0 for every finite natural number i E N,then there is an infinite v 5 p such that xi M 0 for every i 5 v. Most frequently, we shall use the statement for sequences.

PROOF. We cannot use overflow directly on the set of x such that f(x) M 0, because this set is not internal. So we consider instead the set S of all x such that If (y)l 5 l / y for all y 5 x, y in the domain o f f . The set S contains all finite natural numbers larger than a , and, hence, by overflow it contains an infinite number v. Let y 5 v. If y is finite, then f ( y ) M 0, by hypothesis. If y is infinite, then If(y)I 5 l / y , and hence f ( y ) M 0. Thus, f(y) M 0, for all y 5 v M m. The second assertion, for sequences, follows from the first, or it can be proved directly in a similar way. In a similar way, one can prove that for every infinitesimal 6 > 0 there is an infinitesimal E > 0 such that 6 / ~is infinitesimal (or E / S is infinite), i.e., E is a much larger infinitesimal than 6. Also, it is easy to prove that if v is an infinite natural number then there is another infinite natural number p < v such that v - p is infinite.

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VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

If v is an infinite natural number, the set (1, 2, . . . , v} is internal. This fact motivates the following definition:

DEFINITION VIII.3 (HYPERFINITESETS). If the number of elements of an internal set S is v , where v is a natural number, then S is called hyperfinite. More formally, an internal set S is hyperfinite if there is an internal one-one function f from {1,2,. . . ,v} onto S , where v E *N.The number v is called the internal cardinality of S ; we shall use # S to denote the internal cardinality of S . If v, the number of elements of a set S , is an infinite natural number, S is not, strictly speaking, finite, but hyperfinite. In particular, (1, 2, . . . , v} is hyperfinite. Hyperfinite sets are useful, because they can approximate standard sets in a sense we shall discuss later, and, although they may be infinite, by internal induction they behave formally as finite sets; thus combinatorial and counting arguments valid for finite sets are also true for hyperfinite sets. For instance, if T = { t l , t a r . .. , t Y } is a hyperfinite set of numbers (which contains v elements with v x co),then T has a maximal and a minimal element; the union of a hyperfinite family of hyperfinite sets is hyperfinite; the sum of any hyperfinite subset A of *R, CzEAz, always exists. As we shall see later, integration can be obtained from hyperfinite summation. We shall discuss a few examples of how hyperfinite sets can approximate other sets. EXAMPLE VIII.l. Let T = (0, 1 , . . . , v}, where v is an infinite natural number. This is a hyperfinite analogue of N,since the set of standard images of the finite elements of T , st"T, is W. EXAMPLE VIII.2. Consider, now, the closed interval [0, 11. We define the discrete time line T = (0, d t , 2 d t , . . . , l}, where dt = 1/v for a certain infinite v E *N.The set T is clearly internal since it can be defined as the set

( k d t 10 6 k 6 v}.

+

This definition also shows that T is hyperfinite with #T = v 1, because the bijection f ( k ) = k dt is clearly internal. Now, for each real number r in [0, l), there is exactly one k such that k dt 5 r < (6+1) d t . Just take k to be the largest integer 5 v such that k dt 5 r . This k exists by internal induction, since the set of integers 5 Y is hyperfinite (see, also, the observation at the end of page 172). For this k , r - k dt 5 d t , and, hence, r x k d t . Thus, in a sense, T approximates [0,1] fl W,since st"T = [0,11nW. If we take v = p ! for a certain infinite p E *N, then T contains all standard rationals between 0 and 1.

EXAMPLE VIII.3. If instead of taking T the set of k dt for k 5 Y, we consider P = { k d t 10 5 k 5 v2},

3. INTERNAL ANALOGUES OF CONVERGENCE

183

then P approximatm the whole interval [O,+oo), in the sense that for any z E [0, +oo) n R there is a y E P such that y M z,i.e., st” P = [0, +oo) n R. In our development of probability theory, we shall deal almost exclusively with hyperfinite sets and with functions, for instance stochastic processes, from these sets into *R. Important hyperfinite sets are the near intervals as in Example VIII.2:

DEFINITION VIII.4 (NEARINTERVALS). Let v E *N. A hyperfinite set T = { t o , t z , . . . , t Y }is a near interval for [a, b], if t o = u , t , = b, and the difference t j + l - ti = dtj is infinitesimal for every i = 0, 1, 2, . . . , v - 1. As an example, we have that, if T is a near interval for [a,b], then for all M t . It is enough to take the maximum of

x E [u,b] there is a t E T such that x the t E T such that t 5 x.

EXAMPLE VIII.4. We now consider a hyperfinite approximation to the unit circle. We take the unit circle C to be the set of complex numbers 2~ such that 1.1 = 1; that is, the set of numbers eie, for 0 E [0,2?r]. Let v be an infinite natural number, and let d0 = 27r/v. Then a hyperfinite approximation to C is the set

c = { e i k dB 10 5 k 5 v}. Again we have that every element of C is approximated by an element of C. It is clear that C can be obtained from a near interval in the following way. Let

T = (0, d e , 2 d 4 . . . ,2+ Then T is a near interval for [0,27r],and dt = do, for every t E T. We have

c = {eit I t E T}. 3. Internal analogues of convergence

We shall consider three kinds of sequences of numbers: sequences x,, for n E N, sequences z,, for 1 5 n 5 v, where v M 00, and sequences z,,,for n E *M.The first type cannot be.interna1. In order to deal with convergence for this first kind of sequences, we need to add one more principle to our characterization of internal sets: Denumerable comprehension: If 2 1 , 22, . . . , z, . . . , for n E N, is a sequence of elements of an internal set A , then there is an internal sequence of elements of A , 21, x2, . . . , x y , . . . , for v E *N,which extends it. There are several nonstandard analogues of the notion of a convergent sequence. One of them is near convergence of sequences as in [71, p. 201. This analogue uses sequences with domain the numbers less than an infinite natural number. In this case, the sequences contain a hyperfinite number of elements.

184

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

DEFINITION V111.5 (NEARCONVERGENCE). Let v be an infinite natural number. We say that the sequence of hyperreals X I ,22, . . . , x,, is nearly convergent if there is a hyperreal number x such that x p M t for all infinite p 5 v. We also say in this case that the sequence X I , . . . , x,, nearly converges t o 2. We call the usual notion of convergence for sequences with domain W, Sconvergence, that is, the sequence {Z,},~N S-converges to 2, if for every E >> O there is an no E M such that Itn- tI < E for every n E M, n 2 no. Notice that if a sequence 21, . . . , ty nearly converges to x , then it also nearly converges to y if and only if t k! y. The following theorem, which is implicit in the discussion of [71, p. 201, may help to understand why the notion of near convergence is an analogue of S-convergence.

THEOREM VIII.ll. Let 21, z2, . . ., t,, . .., f o r n E MI be a sequence of hyperreal numbers. Then the sequence S-converges t o a number x if and only if there is an internal extension X I ,22, . . ., x,,, f o r a certain u = 00, that nearly converges to 2 . In fact, if the sequence S-converges t o x, then f o r every internal extension X I , x2, . . . , x p ,f o r p x 00, there is a u M 00, u 5 p, such that 21, x2, . . . , x,, nearly converges t o x. PROOF.Suppose that the sequence 2 1 , . .. , t,,nearly converges to x and take the (external) restriction of the sequence to W, i.e., 2 1 , 22,. . . , x,, . . . for n E W. Let E >> 0. The internal set {m I for all n 2 m, 12, - 21 5 E } contains all infinite numbers 5 u. Hence, by overflow, Theorem VIII.7, it contains an n, << 00. Hence, t l , x2, . . . , S-converges to 2. On the other hand, suppose that 21, 22, . . . is a sequence defined on M, S-convergent to t. Then, by denumerable comprehension, we can extend the sequence to an internal sequence on 'W. Let A , = { n E *W I Ix, - X I 5 l/m}, for rn a finite natural number. Then A , is internal and contains all finite n 2 n, for a certain finite n,. Hence, by overflow, Theorem VIII.7, it contains an infinite pmsuch that Ix,-t) 5 l / m for all n with n, 5 n 5 p,. Now, make the sequence p decreasing by taking pA = inf{pj I j 5 m } . Then we still have 12, - t i 5 l/m for n satisfying n, 5 n 5 PA. We now need an infinite number less that p h for every finite rn. Since we cannot take the minimum of all the p;, for all rn E R, bccause it is not internal, we consider an internal extension of the sequence p A to 'M. The internal set S1 = {m E 'M I pi > p h for all j < m} contains all finite natural numbers, and hence by overflow, Theorem VIII.7, it contains an infinite natural number q l . The reciprocals of the pi are infinitesimal for every finite j . Hence, by Robinson's lemma VIII.10, there is an infinite number 92 such that pi is infinite for every j 5 92. Let 9 be the least of 81 and 92 and let u = ph. Then for any finite number m,if n is an infinite number such that n 5 u , then nm 5 n 5 p, and, hence, lxn - 5 l / m . Thus, Itn - x 0, for any infinite n 5 v . Thus, the sequence 21, 22, . . . , x,, nearly converges to x. El

XI

XI

We shall occasionally need another analogue of the ordinary notion of convergence applied to sequences of numbers {z,},~=~.

4. SERIES

185

DEFINITION VIII.6. We say that the sequence {z,,},,~*B *-converges to x, if for every E > 0 (real or not) there is a vo E *N(finite or not) such that for every v 2 vo, It, - 21 < E . We have the following theorem that relates, for standard sequences, the three notions of convergence. For internal sequences that are not standard the theorem which is not true, as it can be seen by the sequence x,, = E M 0, for every n E *N, is internal, nearly converges to zero (and, hence, S-converges to zero), but does not *-converge to zero.

THEOREM VIII.12. Let 2 1 , 22, .. ., x,,,.. . ,for la E *Nbe a standard sequence, and let x E Iw. Then the following conditions art equivalent: (1) The restriction of the sequence to N,{z,,),,Ew, S-converges t o x. ( 2 ) For every v M 00, the restriction o f t h e sequence t o v , {zn),,<,, nearly converges to x. ( 3 ) The sequence *-converges 20 t. PROOF. We have already proved (Theorem VIII.ll) that for internal, and hence standard, sequences (1) and (2) are equivalent. We now prove that (2) implies (3). Suppose (2). Then the standard set A consisting of the E > 0 satisfying that there is an n E 'N such that for all rn 2 n, IZm - 21 5 E contains all positive real numbers. Hence, it contains all positive hyperreals. Thus, we have (3). Finally, assume (3). Let E > 0 be a real number. Then the set B consisting of the n E *Nsuch that for every rn 2 n, 12m - xl 5 E is standard and nonempty. Then, by standard overflow, Theorem VIII.7, B contains a finite n. Thus, we have (1). Hence (3) implies (1). 0 4. Series

We now introduce the nonstandard analogue to convergent series corresponding to convergent sequences. DEFINITION VIII.7 (CONVERGENCE O F SERIES). (1) Let xlI22, . .. , z, be a sequence, where v is an infinite natural number. We say that C:=l x i is nearly convergent if the sequence of partial sums, y,, = C;.'=, xil for n = 1, 2, . . . , v , is nearly convergent. 00 x i is S(2) Let { I , } , ~ B be a sequence. We say that CiEIxi = convergent if the sequence of partial sums, yn = Cy='=,zj, for n = 1, 2, . . . , is S-convergent. (3) Let { z ~ ) ~ E . R Jbe a sequence. We say that CiE.w xj = x i is *convergent if the sequence of partial sums, y,, = Cy='=, x i , for n E *N, is *-convergent.

xEl

The notation Cgl xi is ambiguos. It has a different meaning for a sequence on N and a sequence on *N. Since C:=,xi denotes a number and not a sequence, there is an abuse of x i , always exists, but the series, written language here. Notice that the sum, C:='=l

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

186

in the same way Cr=lxi, does not always nearly converge. In order for the series to nearly converge, we must have Y

P

i=l

for every infinite p 5 v. It is clear that if the series xi nearly converges, then it nearly converges to its sum y = C;='=, xi, and that the series xi nearly converges if and only if the tails J '& xi are infinitesimal for all infinite p 5 v. Thus, we have the following theorem.

cr=l

c;='=,

THEOREM VIII.13 (COMPARISON TEST). Let X I , ..., xu, and y1, .. ., yu be internal sequences where v is infinite. Then if lxnl 5 lgnl for all infinite n 1. v Ixil also nearly converges. and Cr='=, lyil nearly converges, we have that We also have:

THEOREM VIII.14. Let X I , xz, . . . , xu be an internal sequence where v is /xi1 nearly infinite, and suppose that Ixil << 00 for all i << 00 and that converges. Then C:==, Ixil<< 00.

PROOF.The set A of all n such that C:=,]xiI _< 1 contains all infinite numbers. Then by overflow (Theorem VIII.7) it must contain a finite number rn.

But

C

ffl-1

1Xils

( m - I)ma{lxil

I i < m).

i= 1

Hence

is finite.

From Theorem VIII.ll for sequences, we immediately obtain:

THEOREM VIII.15. Let x l , 22, . . ., I,, ..., f o r n E N, be a sequence of hypemeal numbers. Then the series CFl xi S-converges to a number 2 if and only if there is an internal extension X I , 22, . . . , xu, for a certain v M 00, such xi nearly converges to x. that the series In fact, if the sequence S-converges t o x, then for eve y internal extension X I , 22, . . . , xp, for p x 00, there i s a Y ca, v 5 p , such that x; nearly converges to x.

x:=l

A theorem similar to VIII.12 is true for standard series. THEOREM VIII.16. Let X I , 22, . . . , x,,,.. . ,for n E *Nbe a standard sequence. Then the following conditions are equivalent: (1) The restriction of the series, CnEH z,, to M, S-converges to a real number 2.

5. CONVERGmCE OF FUNCTIONS

(2) For every v m 00, the restriction of the series to converges to x. (3) The series Cr=lxn *-coreverges to z.

187 Y,

zn nearly

5. Convergence of functions We need a few facts about internal analogues for the notion of limits at infinity for functions. DEFINITIONvIII.8 (CONVERGENCE A T INFINITY). (1) Let f be a function defined on an interval [a,b] where b M 00 (or a M -00). We say that f @earlyconverges to c at 00 (-00), if f(z)M c for every z m 00, 3: 5 b (z w -00, z 2 u ) . (2) Let f be a function defined on the finite numbers of an interval [a, b] where b M 00 (or a M -00). We say that f S-converges to c at 00 (-co), if for every E >> 0 there is a finite M > 0 ( M < 0) such that, for every - cI < E . finite z with b 2 G > M (u 5 z < M )) . ,( f I (3) Let f be a function defined on an interval [a,00) (or (-00, b]). We say that f *-converges t o c at 00 (-00), if for every E > 0 there is an M > 0 ( M < 0) such that for every z > M (z < M ) , - CI < E .

).(fI

Theorems similar to the theorems on convergence are true for these notions. For instance, we have:

THEOREM VIII.17. LeZj be an internal function defined on an interval [a,b], where b M 00 (u m -00). Then f nearly eonverges to c at infinity if and only if f S-converges t o c at infinity. THEOREM VIII.18. Let f be a standnul function defined on an interval [u,co) ((-co,b]). Then the following conditions are equivalent: (1) The restridion off t o any interval containing the finite numbers in its domain, S-canmerges .at infinity to a real number c. (2) For every b M 00 (a M -00), the restriction off to [a, b] nearly converges at infinity to c. (3) The function f *-converges at infinity to c. The proofs, which are similar to those in the preceding sections, are left to the reader. Finally, we introduce the notion of convergence of functions: OF FUNCTIONS). Let f : A --+ *R. Then DEFINITIONVIII.9 (CONVERGENCE (1) f nearly converges to c at to, if for every t M to, t E A , f ( t ) M c. (2) f S-converges t o c at t o , if for every E >> 0, there is a 6 >> 0 such that It - to1 < 6 , t E A implies .lf(t)- f ( t o ) l < E . (3) f *-converges to c at to7 if for every E > 0, there is a 6 > 0 such that It - to1 < 6 , t E A implies If(t>- f ( t o ) l < E .

We have:

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

188

THEOREM VIII.19. Let f : A + 'R be an internal function. Then f nearly converges t o c at t o if and only iff S-converges to c at to. For standard functions, the three notions of convergence are equivalent:

THEOREM VIII.20. Lei f : A -+ 'R be Q standard function and let to E R. Then the three notions of convergence at t o aw equivalent. The proof are similar to the previous proofk and are left t o the reader.

6. Continuity We now discuss the different notions of continuity, which are analogues of the standard notion of continuity. Let f : A -+ *R, where A 5 n*R; then DEFINITIONVIII.10 (CONTINUITY). (1) f is nearly continuous at k E A if whenever y" i? with y" E A , we have f(y3 R f(Z). f is nearly continuous on A if it is nearly continuous at each k in A. (2) We say that f is S-continuous at 2 E A , if for every E >> 0 there is a 6 >> 0 such that if -1'. 41 < b and y ' A~, we have that If (Z) - f($/ < E . f is S-continuous on A if it is S-continuous at each 2 in A . (3) Finally, we say that f is *-continuous at 2 E A , if for every E > 0 there is < E. a 6 > OsuchthatifI/Z-dI < b a n d y ' € A , wehavethat If(Z)-f($I f is *-continuous on A if it is *-continuous at each Z in A . We shall use the notion of near continuity mainly when A is a near interval T. But the notion makes sense also for other types of sets. In order to see that near continuity at t is a nonstandard analogue of continuity at t , we have the following theorem.

THEOREM VIII.21. Lei f : A -+ *R, with A C n*R, be internal. Then f is nearly continuous at i? E A if and only iff as S-continuous at 2. PROOF. Let f be internal nearly continuous at i? E A and let E >> 0. Let A3 be the set of all 6 such that for all y' E A , if Ily"-- ql 5 6 then lf(y3 - f (k)l5 E . Since the internal set LIZ contains all positive infinitesimals, by overflow (Theorem VI11.7) it contains a b >> 0. Thus, f is S-continuous at Z. Suppose, now, that f is S-continuous at k. Then for every E >> 0 there is a 6 > 0 such that if y'E A and \If- 511 < 6, then lf(y3 - f ( Z ) l < E . Let G E A , y ' 2~and let E >> 0. Then we have that IIy'< 6, for the 6 that works for E , because 6 > 0. So that, since E >> 0 was arbitrary, I f (9- f(Z)1 < E , for any E >> 0. Thus, If(y3 - !().'I e 0. Therefore, f is nearly continuous at Z . 0 Now, if T is hyperfinite then near continuity on T is the internal analogue of uniform continuity, since we have:

THEOREM VIII.22. Let T 5 "*Pi be hyperfinite. Then the internal fanction *R is nearly continuous on T if and only if f o r every E >> 0 there is a 6 >> 0 such that f o r any ( s'E T with - ;1 Sll 5 6 we have - f ( i ' > I 5 E. f

;T

-+

If(;)

7. DIFFERENTIALS AND DERIVATIVES

PROOF.Let f : T + 'JR be nearly continuous on T and let ;E T, let ATbe as in the previous proof, and let

189 E

> 0.

For each

1 A = {TI E *W I - E A?}. n Then A is an internal subset of 'N, and, thus, by internal induction, it has a least element n. Let 1/n = 61, i.e., 6~ is the largest element in AT that is the reciprocal of an integer. Then, as we saw above, 0 << 6 ~ .Let

6 = min(6,- I > 0, there is a 6 >> 0, such that if if,
THEOREM VIII.23. Let f be

a standard function defined on A C "'R Then the following conditions are equivalent: (1) f is nearly continuous at &. (2) f is S-continuous at &. (3) f is *-continuous at &.

& E A , ;real.

and

PROOF.By Theorem VIII.21, we know that (1) and (2) are equivalent. So, assume (2). The set B of all E > 0 for which there is a 6 > 0 such that for all < E A with - ;1 611 < 6 we have I f ( t )- f ( t o ) l < E is standard and includes all positive reals. Then B includes all positive hyperreals. Thus, we have (3). Finally, assume (3) and let E > 0 be real. The set B of all 6 > 0 such that for < 6 we have If(t) - f ( t o ) l < E is standard and nonempty. all
We need to define when a point is in the interior of a set:

VIII.ll. We say that a' is in the interior of a subset A of n*R, if DEFINITION I E A , for every I =2. We now define differentials as differences: DEFINITION VIII.12 (DIFFERENTIALS). (1) We take dx to be a nonzero infinitesimal. Then, for any real function of one variable f such that x and x dx are in its domain, we define

+

df(x, dx) = f ( x

+ dx) - f ( x ) .

190

VIII. ELEMENTS O F INFINITESIMAL ANALYSIS

(2) The function f is diferentiable at a point function g , independent of d x , such that

d f ( x ,d x ) M g ( 4 dx for every infinitesimal dx and x domain of f.

M

Q

of its domain if there is a

(dx),

a , such that x and x

+ dx are in the

This is the traditional definition of differentials that was adopted in 1781, but not in other versions of nonstandard calculus such as [51] and [52]. Notice that the definition of differentiable at a point a is equivalent to saying

for every infinitesimal dx and x M a . Recall that the derivative of a standard function f at a red point t o is the limit of the differential quotient

f(t0

+ d t ) - f( t o ) dt

when dt tends to zero. We say that a function f is derivable at an interior point t o if the derivative exists at to. We see, from Theorem VIII.20, that if a standard function f is differentiable, then it is derivable, and

for every infinitesimal dt and every t % t o . Also, it can be proved in this case that there is exactly one standard function f' which is the derivative of f . We shall call this standard function, f', the derivative o f f . It is not difficult to show that this definition coincides with the usual standard definition. We proceed, now, to introduce partial differentials and derivatives for functions of several variables. For simplicity of expression, we define these notions for functions of two variables. The extension to n variables is easy.

DEFINITION VIII.13 (PARTIAL DIFFERENTIALS). Let z = F(z,y) and (x,y) be in the domain of F . (1) If we let y be constant, we define

+

d , ~= ~ , F ( xy), = F(z d z , y) - F ( z , y), for ( x + d x , y) in the domain of F . Similarly, letting x be constant 892

+

= ~ ~ F ( x ,=YF) ( x , Y + d y ) - F ( ~ , Y ) ,

for (2,y d y ) in the domain of F . The functions 8,z and 3,z are called partial differentials.

7. DIFFERENTIALS AND DERIVATIVES

(2) If there is a function

-

191

F,, independent of dx, such that

ax%

Fx(x, 9) dx

(w,

for every infinitesimal dx with ( x + d x , y ) in the domain of F, then F,(x,y) is called a partial derivative of F with respect t o x at (x,y). Similarly, we define Fy(x, y). The definition of partial derivative is equivalent to a,z

where

E B 0.

= F,(x,y)dx+~dx

That is

for every dx B 0. Suppose, now, that F is standard. Define for x, y real

and take the standard extension, called also F,. This is t h e derivative of a standard function, which can be shown to be unique. Alternative notations for the partial derivatives are

Similarly as for functions of one variable, these definitions of partial derivatives are equivalent, for standard functions, to the standard definitions. DIFFERENTIAL). Let z = F ( z , y) and. (z,y) be DEFINITIONVIII.14 (TOTAL in the domain of F. Then:

(1) The total diferential or, simply diferential dz = dF(2, y) = F ( z

oft

= F ( x , y) is

+ AX,y + Ay) - F ( z ,Y),

+

where (x + Ax, y Ay) is in the domain of F . The function dz is, really, a function of four variablks and should be written d F ( z , y, Ax, Ay). (2) For standard functions F , the total derivative or, simply dtrivative of z = F(x, y) is defined by

Dz = D F ( x , y) = Fx(x, y) Ax

Again, D z is a function of four variables.

+ Fy(z,Y)A Y

192

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

Although we shall use d F and DF mainly for Ax and Ay infinitesimal, the definition makes sense for any hyperreal number, and, for some proofs, it is necessary to have the functions defined for all hyperreal numbers. We write dx and dy in case we are assuming that they are infinitesimal. For instance, if t = xy

and

Dz = ydx

+ xdy.

The length of the vector (dx, dy) will be written ds. That is ds

= J(dz)2

+ (dy)2.

If we don’t assume that dx and dy are infinitesimal, then we write

AS = J-. DEFINITION VIII.15. We say that the function F is differentiable at a point

(zo,y~)in its domain, if for every infinitesimal vector (dx,dy) such that the

domain of F contains (20

+ dx,yo + dy)

dF(xo, YO)M DF(zo, YO)

(ds).

The last expression can also be written

dt

M

Dz

(ds).

The differentiability of F is, of course, equivalent to

+ E ds

dF(xo, YO)= DF(xo, YO)

with E M 0. It is not difficult to show, that for standard functions, the notion of differentiability we have defined is equivalent to the standard notion. That is:

THEOREM VIII.24. Let F be a standardfunction and (20, yo) be a real interior point in its domain. Then the following conditions are equivalent: (1) For every dx and dy injinitesimals, there are €1, ~2 infinitesimal, such that dF(X0, Yo) = DF(X0, Yo) El dx € 2 dY. (2) F is diflerentiable at ( X O , yo). (3) For every E >> 0 , there is a 6 >> 0, such that, Z . f ~ ~ ( A x , A y<) 6~,~then

+

F(xo

+

+ Ax, Yo + AY) - F(X0,YO) - DF(X0, Yo)

Jm

I

< E.

7. DIFFERENTIALS AND DERIVATIVES

193

(4) F o r every E > 0 , there is a 6 > 0 such that, if \ \ ( A x ,A y ) l \ < 6 , then

PROOF. Notice, first, that if d x

> 0, then

d x-

1

cis and if d x

<0

dx

1

Thus, d x l d s is always finite, and if d x 2 d y , d x l d s is not infinitesimal. The same is true for d y l d s . We shall show, first, that (1) is equivalent to (2). Assume (1). We have

+ 61 d x + ~2 d y ,

dz = DZ for certain

~1 M 0 M € 2 .

Dividing both sides b y d s and rearranging

dz ds

DZ dx = €1ds ds

---

+ E 2 -dd ys M 0.

Thus, we have (2). Assume that (1) is not true for certain infinitesimal d x and dy. Suppose, also, that d x 2 dy. (The proof for the case d x 5 d y is similar.) Let 6 2 M 0. Then, since (1) does not hold d r - DZ dy - &adx dx is not infinitesimal. Because d x 2 d y , d y l d x is finite and, hence, ~ ~ ( d y l dMx )0. Thus d z - DZ dx is not infinitesimal. We have d z - DZ d z - DZ d x ds = d x ds/'

I

I 1

As we saw above, since d x 2 d y , d x l d s is not infinitesimal. Therefore

d z - DZ ds is not infinitesimal. Thus we have shown the negation of (2). Hence, we have completed the proof of the equivalence between (1) and (2). Assume, now, (2) in order to prove (3). Let 6 such that

E

> 0, real.

Then the set A of the

VIII. ELEMENTS OF INFINITESIMAL ANALYSIS

194

is standard and contains all positive infinitesimals, hence, by standard overflow, Theorem VIII.7, it contains a certain 6 > 0 real. With this we have (3). Assume (3). The standard set B of the E such that there is a 6 satisfying

contains all positive reals, and hence, all positive hyperreals. Thus, we have shown (4). Finally, assume the negation of (2). Let ds be such that dz - DZ ds is not infinitesimal, with ds M 0. Thus

for certain

E

> 0 real. The standard set

C = (6 I there is a As with IAsI < 6 and

1

d z - DZ

As

contains all positive reals and, hence, all positive hyperreals. Therefore, we have shown the negation of (4). 0 An almost immediate corollary of the definition of differentiability is:

COROLLARY VI11.25. If F is diflerentiable at a point ( x 0 , y o ) in its domain, then F is nearly continzlous at (z0,yo). PROOF.Assume that with

E M

dr = D z

+ E ds

0. Then F(zo+dz,yo+dy) = F ( ~ o , y o ) + D z + ~ d s .

But both Dz and E d s are infinitesimals. Therefore q.0

+ d x , Yo + d Y ) M F ( z 0 , Yo).

0

DEFINITION VIII.16 (SMOOTH FUNCTIONS). We call a standard function F smooth at an interior point (z0,yO) of its domain, if both partial derivatives exist and are continuous at (to, yo). We shall not prove the Mean Value Theorem for the differential calculus, but

we assume it for standard functions.

THEOREM VIII.26. Assume that z = F ( x , y ) is smooth (and standard) at an interior point (zo, yo) of its domain. Then F is differentiable at ( 2 0 ,yo).

7. DIFFERENTIALS AND DERIVATIVES

195

PROOF. We shall show that

dz = D Z+ ~1 dz + ~2 dy

at

(20, yo) with ~1 and €2 infinitesimals. Assume, first, that (z0,yo) is real. We have

It is clear that

F(.o

+ dz, Yo) - F ( z 0 ,Yo) = a d =Fs(zo,yo)dz

(2) with ~1 M 0. We shall show

+

+

EI

dz

+

+

+

F(zo d z , YO dy) - F ( z o dz, yo) = Fy(zo,YO) dy ~2 Cay (3) with 6 2 M 0. As (20 d z , y ) M ( z ~ , y ) ,Fv(zO dz, y) is continuous at every y between yo and yo dy, by the Mean Value Theorem

+

+

+

+ dz, 90 + 4 4 ) - F(.o + d z , Yo) = Fy(z0 + d z , Y1) dY for a certain y1 between yo and yo + dy. Since Fy is continuous at (10, yo) and F(.o

Y1 M Y 0

with

Fy(+o+dz,Yl)= Fy(t0,Yo) + E 2

~2 M 0. q.0

Thus

+ dz, Yo + dY) - F ( z o + dz, YO) = (Fy(z0,Yo) +

.2)

dY,

which is equivalent to (3). Replacing (2) and (3) in (l),we obtain the theorem for (20, yo) real. Because the condition of differentiability is standard, we obtain the theorem for all hyperreal vectors. 0 The generalization to functions of n variables is straightforward. Let w = F ( z 1 , 22,. . . ,z,) be a function of n variables. Then, the differential d w is

d~ = F(zl+

d . 1 ,

t2

+ d 2 2 , . . . ,2, + dzn) - F(z1,~ 2 , . .. ,t n )

and the total derivative

for dz1, dz2, . . . , dz, infinitesimals. The same theorems are valid with similar proofs. We also shall have occasion to use the Implicit Function Theorem for standard functions, which we assume proved. We also assume the Chain Rule.