By Robert Dalmasso

Gasquet C., Witomski P. examine de Fourier et purposes (Dunod, )(fr)(ISBN 2100050176)(C)(366s)

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**Example text**

B) Wheneverx,y EX and x contained in X. 1}. Proof First we show that (a) implies (b). 4· Continuity and connectedness 431 find points x < y in X such that [x, y] is not contained in X. Then there exists a real number x < z < y such that z ¢ X. Thus the sets (-oo, z) n X and (z, oo) n X will cover X. But these sets are non-empty (because they contain x and y respectively) and are open relative to X, and so, X is disconnected, a contradiction. Now we show that (b) implies (a). Let X be a set obeying the property {b).

Define the l 1 and metrics on this space by 00 dtl((an):=O• (bn):=o) := L n=O ian- bnl; zoo 400 12. Metric spaces Show that these are both metrics on X, but show that there exist sequences x< 1>, x< 2 >, ... , sequences of sequences) which are convergent with respect to the d1oo metric but not with respect to the d11 metric. Conversely, show that any sequence which converges in the dp metric automatically converges in the d1oo metric. 16. Let (xn)~=l and (Yn)~=l be two sequences in a metric space (X,d).

Since E = VnY, this means that B(X,d)(x, r)nY is contained in E. ), and so B(Y,dJyxy)(x,r) is contained in E. Thus x is an interior point of E in the metric space (Y, diYxY), as desired. 1. 4(b). 4 Cauchy sequences and complete metric spaces We now generalize much of the theory of limits of sequences from Chapter 6 to the setting of general metric spaces. 1 (Subsequences). Suppose that (x(n))~=m is a sequence of points in a metric space (X, d). Suppose that n1, n2, n3, ... 2. the sequence ((p, p))~ 1 in R 2 is a subsequence of the sequence ((~, ~))~=l (in this case, nj := j 2).