n=3. Suppose that (X, d) is a metric space which is not compact. Let {an}_1 be a sequenceof distinct points in X that has no convergent subsequences. For each n€ N, take˜n > 0 such that Brn (an) Brm (am) = Ø) for n ‡ m. Letfn(2):=andrn - d(x, a)™n + d(x, an)nf(x) = { fn(x), if a € Br. (an) for some n € N;otherwise.9Show that fn and f are continuous functions. Is f bounded?

Answered: Image /qna-images/answer/f7dd07fc-e400-406a-9d23-3a1c2a1374d4.jpg

3. Suppose that (X, d) is a metric space which is not compact. Let {an}_1 be a sequence of distinct points in X that has no convergent subsequences. For each nЄ N, take rn > 0 such that Brn (an)Brm (am) = 0 for n‡m. Let and fn(2): := rn - d(x, a) ™n +d(x, an) n I fn(x), if x € Brn (an) for some n € N; otherwise. f(x) = { 0, Show that fn and f are continuous functions. Is f bounded?

3. Suppose that (X, d) is a metric space which is not compact. Let {an}_1 be a sequence of distinct points in X that has no convergent subsequences. For each nЄ N, take rn > 0 such that Brn (an)Brm (am) = 0 for n‡m. Let and fn(2): := rn - d(x, a) ™n +d(x, an) n I fn(x), if x € Brn (an) for some n € N; otherwise. f(x) = { 0, Show that fn and f are continuous functions. Is f bounded?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 23E

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