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 (n)-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 € X.
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 (n)-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 € X.
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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Transcribed Image Text:Let (fn) 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 € X.
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