Suppose {fn}n=1 and {gn}n=1 converges uniformly on E where {fn}n=1 and {gn}n=1 are sequences of bounded functions. Prove that {fgn}n=1 converges uniformly on E.
Suppose {fn}n=1 and {gn}n=1 converges uniformly on E where {fn}n=1 and {gn}n=1 are sequences of bounded functions. Prove that {fgn}n=1 converges uniformly on E.
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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Question 5
![4. Consider the sequence of functions {f}-2 where fn: [a, b] R defined for n > 2 by
if 0 sxs
fn = {-n² (x-) if <x<.
if <x<1
n²x
72
0.
a. Prove {fn}2 converges pointwise to f(x) = 0.
b. Prove {fn}-2 does not converge uniformly to f(x) = 0.
Show lim f (x)dx + lim A(x)dx by computing each integral where
C.
72- CO
graphing fn(x) may help.
5. Suppose {fn}n=1 and {gn-1 converges uniformly on E where {f-1 and {g,}-1 are
sequences of bounded functions. Prove that {fgn}, converges uniformly on E](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfc5bdab-6654-444e-b9d5-5fc4b8470fbb%2Fdebe7c9f-f163-4e76-b4e9-c10c0fd1b288%2Fk9du0q8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Consider the sequence of functions {f}-2 where fn: [a, b] R defined for n > 2 by
if 0 sxs
fn = {-n² (x-) if <x<.
if <x<1
n²x
72
0.
a. Prove {fn}2 converges pointwise to f(x) = 0.
b. Prove {fn}-2 does not converge uniformly to f(x) = 0.
Show lim f (x)dx + lim A(x)dx by computing each integral where
C.
72- CO
graphing fn(x) may help.
5. Suppose {fn}n=1 and {gn-1 converges uniformly on E where {f-1 and {g,}-1 are
sequences of bounded functions. Prove that {fgn}, converges uniformly on E
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