3. For each n€ Z+ let fn : [0, 1] → R, fn(x) = 1/(1+x").(a) Prove that (fn) converges uniformly.1/2(b) Compute the limit limn→XS"fn, rigorously justifying your answer.

We have solved both the parts. I hope it will help you.Explanation:### Part (a): Prove that (fn)…

Answered: 3. For each n€ Z+ let fn : [0, 1] → R,…

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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3. For each n € Z+ let fn: [0, 1] → R, fn(x) = 1/(1+x"). (a) Prove that (fn) converges uniformly. (b) Compute the limit lim -1/2 fn, rigorously justifying your answer.
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#1. Thanks.
1. Let f : (0,1) → R be uniformly continuous. Let (xn) be a sequence in (0, 1). Suppose that lim,, 0 Tn = 0. Prove that lim, >∞ f (Tn) exists.

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3. For each n€ Z+ let fn : [0, 1] → R, fn(x) = 1/(1+x"). (a) Prove that (fn) converges uniformly. 1/2 (b) Compute the limit lim n→X S" fn, rigorously justifying your answer.