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only if 3. Let X and Y be metric spaces. Let f : X → Y be a function. Let (a;) be a Cauchy sequence in X. (a) Prove that if ƒ is uniformly continuous, then (f(a;)) is a Cauchy sequence in Y

only if 3. Let X and Y be metric spaces. Let f : X → Y be a function. Let (a;) be a Cauchy sequence in X. (a) Prove that if ƒ is uniformly continuous, then (f(a;)) is a Cauchy sequence in Y

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Real Analysis 

only if lim diameter an 3. Let X and Y be metric spaces. Let f: X →Y be a function. Let (a,) be a Cauchy sequence in X. (a) Prove that if ƒ is uniformly continuous, then (f(a;)) is a Cauchy sequence in Y. (b) Give an example of a continuous functionf and a Cauchy sequence (a;) that shows that if f is not uniformly continuous, then (f(a)) need not be a Cauchy sequence. [Hint: You need not seek anything very "pathological." Try working with functions f : (0, 1) – R.)
Transcribed Image Text:only if lim diameter an 3. Let X and Y be metric spaces. Let f: X →Y be a function. Let (a,) be a Cauchy sequence in X. (a) Prove that if ƒ is uniformly continuous, then (f(a;)) is a Cauchy sequence in Y. (b) Give an example of a continuous functionf and a Cauchy sequence (a;) that shows that if f is not uniformly continuous, then (f(a)) need not be a Cauchy sequence. [Hint: You need not seek anything very "pathological." Try working with functions f : (0, 1) – R.)
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