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.
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.
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 (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 on
X, must the convergence be uniform? Justify.
Note: The function h : X → R is bounded if and only if there
exists M > 0 such that |h(x)| ≤ M for all ¤ ¤ X.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbab873e3-c9d9-439d-9721-884c57f8a4bc%2F6fe885c3-a1d2-4259-b056-f7feb87e8344%2Fyrfg3ft_processed.png&w=3840&q=75)
Transcribed Image Text: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 on
X, must the convergence be uniform? Justify.
Note: The function h : X → R is bounded if and only if there
exists M > 0 such that |h(x)| ≤ M for all ¤ ¤ X.
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