The third edition of this well known text continues to provide a solid foundation in mathematical analysis for undergraduate and first-year graduate students. The text begins with a discussion of the real number system as a complete ordered field. (Dedekind's construction is now treated in an appendix to Chapter I.) The topological background needed for the development of convergence, continuity, differentiation and integration is provided in Chapter 2. There is a new section on the gamma function, and many new and interesting exercises are included.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.

a n n ; > n ; _ 1 such that d(x, Pn ) < 2 - i c5. Thus . • • • _ d(q, Pn) < 2 1 - ib for i = 1 , 2, 3, . . . . This says that {pn J converges to q. Hence q E £*. C A UC H Y S E Q U E NC E S Definition A sequence {pn} i n a metric space X i s said to be a Cauchy sequence if for every e > 0 there is an integer N such that d(pn , Pm) < e if n > N and m > N. In our discussion of Cauchy sequences , as well as in other situations which wi ll arise later, the following geomet ric concept wi l l

which evidently diverges. In many cases which occur in applications, the terms of the series decrease monotonically. The following theorem of Cauchy is therefore of particular interest. The striking feature of the theorem is that a rather "thin" ' subsequence of {an} determines the convergence or divergence of ran . Suppose a 1 > a 2 > a 3 > · · · > 0. Then the series 2::- t an con verges if and only if the series 3.27 Theorem 00 L 2"a 2 " = a 1 + 2a 2 + 4a4 + 8a8 + · · · k=O (7)

for each point p of X, a number {) > 0 having the property specified in Defi niti on 4. 5 . This {J depends on e and on p. Iff is, however, uniformly continuous on X, then it is possible, for each e > 0, to find one number 1J > 0 which wi l l do for all poi nts p of X. Evidently, every uniformly continuous functi on is continuous. That the two concepts are equivalent on compact sets follows from the next theorem . CONTI N U ITY 4.19 91 Theorem Let f be a continuous n1apping o.f a co1npact

X;), n U(P, f) = L M ; Ax; , i= 1 L(P, f) = L m ; Ax ; , i= 1 n and finally (1) Ia f dx = inf U(P, f), (2) r f dx = sup L( P, f), -b _a where the inf and the sup are taken over all partitions P of [a, b] . The left members of (1 ) and (2) are called the upper and lower Riemann integrals of f over [a, b ], respectively. I f the upper and lower integrals are equal, we say that f is Riemann integrable on [a, b ], we write f e §I (that is, � denotes the set of Riemann integrable

oo such that l f,(p;) I < M ; for all n. If M = max ( Mh . . . , M,), then l f(x) I < M + e for every x e K. This proves (a). (b) Let E be a countable dense subset of K. ( For the existence of s uch a set £, see Exercise 25 , Chap. 2. ) Theorem 7.23 shows that {.fn} has a subsequence {.fnJ such that {fnlx)} converges for every x e £. Put f,. , = g ; , to simplify the notation. We shall prove that {g ; } converges uniformly on K. Let e > 0, and pick <5 > 0 as i n the begi nning of this proof. Let