Advanced Calculus of Several Variables (1973)
APPENDIX
The Completeness of
In this appendix we give a self-contained treatment of the various consequences of the completeness of the real number system that are used in the text.
We start with the least upper bound axiom, regarding it as the basic completeness property of . The set S of real numbers is bounded above (respectively below) if there exists a number c such that (respectively ) for all . The number c is then called an upper (respectively lower) bound for S. The set S is bounded if it is both bounded above and bounded below.
A least upper (respectively greatest lower) bound for S is an upper bound (respectively lower bound) b such that (respectively ) for every upper (respectively lower) bound c for S. We can now state our axiom.
Least Upper Bound Axiom If the set S of real numbers is bounded above, then it has a least upper bound.
By consideration of the set , it follows easily that, if S is bounded below, then S has a greatest lower bound.
The sequence of real numbers is called nondecreasing (respectively nonincreasing) if (respectively ) for each n 1. We call a sequence monotone if it is either nonincreasing or nondecreasing. The following theorem gives the “bounded monotone sequence property” of .
Theorem A.1 Every bounded monotone sequence of real numbers converges.
PROOF If, for example, the sequence is bounded and nondecreasing, and a is the least upper bound for S that is provided by the axiom, then it follows immediately from the definitions that limn→∞ xn = a.
The following theorem gives the “nested interval property” of .
Theorem A.2 If is a sequence of closed intervals (In = [an, bn]) such that
(i) for each n 1, and
(ii) limn→∞(bn − an) = 0,
then there exists precisely one number c such that for each n, so c = .
PROOF It is clear from (ii) that there is at most one such number c. The sequence is bounded and nondecreasing, while the sequence is bounded and nonincreasing. Therefore a = limn→∞ an and b = limn→∞ bn exist by Theorem A.1. Since an bn for each n 1, it follows easily that a b. But then (ii) implies that a = b. Clearly this common value is a number belonging to each of the intervals .
We can now prove the intermediate value theorem.
Theorem A.3 If the function f : [a, b] → is continuous and f(a) < 0 < f(b), then there exists c (a,b)such that f(c) = 0.
PROOF Let I1 = [a, b]. Having defined In, let In+1 denote that closed half-interval of In such that f(x) is positive at one endpoint of In+1 and negative at the other. Then the sequence satisfies the hypotheses of Theorem A.2. If c= , then the continuity of f implies that f(c) can be neither positive nor negative (why?), so f(c) = 0.
Lemma A.4 If the function f : [a, b] → is continuous, then f is bounded on [a, b].
PROOF Supposing to the contrary that f is not bounded on I1 = [a, b], let I2 be a closed half-interval of I1 on which f is not bounded. In general, let In+1 be a closed half-interval of In on which f is not bounded.
If c = , then, by continuity, there exists a neighborhood U of c such that f is bounded on U (why?). But if n is sufficiently large. This contradiction proves that f is bounded on [a, b].
We can now prove the maximum value theorem.
Theorem A.5 If f: [a, b] → is continuous, then there exists c [a, b] such that f(x) f(c) for all x [a, b].
PROOF The set f([a, b]) is bounded by Lemma A.4, so let M be its least upper bound; that is, M is the least upper bound of f(x) on [a, b]. Then the least upper bound of f(x) on at least one of the two closed half-intervals of [a, b] is also M; denote it by I1. Given In, let In+1 be a closed half-interval of In on which the least upper bound of f(x) is M. If c = , then it follows easily from the continuity of f that f(c) = M.
As a final application of the “method of bisection,” we establish the “Bolzano–Weierstrass property” of .
Theorem A.6 Every bounded, infinite subset S of has a limit point.
PROOF If I0 is a closed interval containing S, denote by I1 one of the closed half-intervals of I0 that contains infinitely many points of S. Continuing in this way, we define a nested sequence of intervals {In}, each of which contains infinitely many points of S. If c = , then it is clear that c is a limit point of S.
We now work toward the proof that a sequence of real numbers converges if and only if it is a Cauchy sequence. The sequence is called a Cauchy sequence if, given > 0, there exists N such that
It follows immediately from the triangle inequality that every convergent sequence is a Cauchy sequence.
Lemma A.7 Every bounded sequence of real numbers has a convergent subsequence.
PROOF If the sequence contains only finitely many distinct points, the the conclusion is trivial and obvious. Otherwise we are dealing with a bounded infinite set, to which the Bolzano–Weierstrass theorem applies, giving us a limit point a. If, for each integer k 1, ank is a point of the sequence such that , then it is clear that is a convergent subsequence.
Theorem A.8 Every Cauchy sequence of real numbers converges.
PROOF Given a Cauchy sequence , choose N such that
Then if n N, so it follows that the sequence is bounded. By Lemma A.7 it therefore has convergent subsequence .
If , we want to prove that limn→∞ an = a. Given > 0, choose M such that
Then choose K such that nk M and . Then
as desired.
The sequence of points in k is called a Cauchy sequence if, given > 0, there exists N such that
(either Euclidean or sup norm). It follows easily, by coordinatewise application of Theorem A.8, that every Cauchy sequence of points in k converges.
Suggested Reading
One goal of this book is to motivate the serious student to go deeper into the topics introduced here. We therefore provide some suggestions for further reading in the references listed below.
For the student who wants to take another look at single-variable calculus, we recommend the excellent introductory texts by Kitchen [8] and Spivak [17]. Also several introductory analysis books, such as those by Lang [9], Rosenlicht [13], and Smith [16], begin with a review of single-variable calculus from a more advanced viewpoint.
Courant's treatment of multivariable calculus [3] is rich in applications, intuitive insight and geometric flavor, and is my favorite among the older advanced calculus books.
Cartan [1], Dieudonné [5], Lang [9], and Loomis and Sternberg [10] all deal with differential calculus in the context of normed vector spaces. Dieudonne‘s text is a classic in this area; Cartan’s treatment of the calculus of normed vector spaces is similar but easier to read. Both are written on a considerably more abstract and advanced level than this book.
Milnor [11] gives a beautiful exposition of the application of the inverse function theorem to establish such results as the fundamental theorem of algebra (every polynomial has a complex root) and the Brouwer fixed point theorem (every continuous mapping of the n-ball into itself has a fixed point).
The method of successive approximations, by which we proved the inverse and implicit function theorems in Chapter III, also provides the best approach to the basic existence and uniqueness theorems for differential equations. For this see the chapters on differential equations in Cartan [1], Lang [9], Loomis and Sternberg [10], and Rosenlicht [13].
Sections 2 and 5 of Chapter IV were influenced by the chapter on multiple (Riemann) integrals in Lang [9]. Smith [16] gives a very readable undergraduate-level exposition of Lebesgue integration in n. In Section IV.6 we stopped just short of substantial applications of improper integrals. As an example we recommend the elegant little book on Fourier series and integrals by Seeley [14].
For an excellent discussion of surface area, see the chapter on this topic in Smith [16]. Cartan [2] and Spivak [18] give more advanced treatments of differential forms; in particular, Spivak‘s book is a superb exposition of an alternative approach to Stokes’ theorem. Flanders [6] discusses a wide range of applications of differential forms to geometry and physics. The best recent book on elementary differential geometry is that of O'Neill [12]; it employs differential forms on about the same level as in this book. Our discussion of closed and exact forms in Section V.8 is a starting point for the algebraic topology of manifolds—see Singer and Thorpe [15] for an introduction.
Our treatment of the calculus of variations was influenced by the chapter on this topic in Cartan [2]. For an excellent summary of the classical applications see the chapters on the calculus of variations in Courant [3] and Courant and Hilbert [4]. For a detailed study of the calculus of variations we recommend Gel'fand and Fomin [7].