Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 171The proof, which is similar to that of Problem 10(c) and (d) following Section 2.4,is left as an exercise (Problem 3).A set G in EN is said to be bounded if 1x1 ( K for some constant K, for allx = (xl, . . . , x,) in G. This is again consistent with the definition for N = 1 ofbounded sets of real numbers.THEOREM E (Weierstrass-Bolzano Theorem for EN) Let P, be a sequenceof points in the bounded set G of EN. Then Pn has a convergent subsequence.Proof. For simplicity we take N = 2, so that P, = (x: , x,") = xn. Since Ixn 1 5 K,we conclude from Rule (iv) that Ix; I ( K, 1x2" I 5 K for all n. Thus x; is a boundedsequence of real numbers. By Theorem C it has a convergent subsequence x;\ Thesequenceis a subsequence of Pn, and x,"' is a bounded sequence of real numbers. By TheoremC again, this sequence has a convergent subsequence. By relabeling we canagain denote this by x,"" it is still a subsequence of xi, and the corresponding sequencex;"~ a subsequence of a convergent sequence and therefore also converges(Problem 2 below). Therefore by Theorem D the new sequence P,, = (x;" xx,"")converges and is the desired subsequence of Pn.THEOREM F A set G in E~ is closed if and only if, for each convergent sequenceP,, in G, the limit of the sequence is in G.For the proof, see Problem 12 following Section 2.4.We consider a mapping f from a set G in EN into EM. This is said to be continuousat Po in G if for each E > 0, there is a 6 > 0 such that d( f (P), f (Po)) < rwhenever d(P, Po) < 6. The mapping f is said to be continuous in G if it is continuousat every point of G.THEOREM G Let f be a mapping of a set G in EN into E ~ Then . f is continuousat Po in G if and only if f (P,) + f (Po) for every sequence Pn in G converging toPo.Proof. Let f be continuous at Po and let Pn + Po, with PI, P2, . . . all in G. GivenE > 0, we choose 6 > 0 as in the definition of continuity above. Then we choosen, sothatd(P,, Po) < S forn > n,.Thend(f(Pn), f(Po)) < E forn > n,, so thatf (Pn) + f (PO).Conversely, let f be such that f (P,) + f (Po) wherever P, + Po. Suppose fis not continuous at Po. Then for some E > 0 for every 6 > 0 there is some P inG such that d(P, Po) < 6 but d( f (P), f (Po)) > c. We take 6 successively equalto 1/2,1/4, . . . , 1/2", . . . to obtain a sequence Pn such that d(P,, Po) < 2-" andd( f (P,), f (Po)) > E. Then clearly, P, + Po, but f (P,) does not converge to f (Po),contrary to assumption. Hence f must be continuous at Po.THEOREM H Let G be a bounded closed set in EN. Let f be a continuousmapping of G into EM. Then the range of f is also bounded and closed.