From Classical to Modern Analysis by Rinaldo B. Schinazi

Author:Rinaldo B. Schinazi
Language: eng
Format: epub, pdf
ISBN: 9783319945835
Publisher: Springer International Publishing

For every n, the function f n is continuous on [0, 2] (why?). By definition,

Hence,

By the Archimedean property there exists a natural N such that . Therefore, for n ≥ N and p ≥ N,

Hence, (f n) is Cauchy in (C([0, 2]), d r).

We now show that (f n) does not converge in (C([0, 2]), d r). By contradiction assume that (f n) does converge to some f in this metric space. We need f to be continuous on [0, 2] and d(f n, f) to converge to 0. We have

Since both terms are positive in the r.h.s. we need both terms to converge to 0. Note that is a constant so it has to be equal to 0. Since |1 − f| is continuous on [1, 2] (why?) and its integral is 0 the function |1 − f| is identically 0 on [1, 2], see Lemma 1.1. Hence,

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