1. Let (fn)1 be a sequence of bounded real-valued functions on X.(a) If fn ⇒ f on X, show that f is bounded on X.(b) If (fn)1 converges pointwise to a bounded function f onX, must the convergence be uniform? Justify.Note: The function h : X → R is bounded if and only if thereexists M > 0 such that |h(x)| ≤ M for all ¤ ¤ X.

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Answered: 1. Let (n)-1 be a sequence of bounded…

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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1. Let (n)-1 be a sequence of bounded real-valued functions on X. (a) If ƒn ⇒ ƒ on X, show that ƒ is bounded on X. (b) If (fn) 1 converges pointwise to a bounded function f on X, must the convergence be uniform? Justify.