Suppose fn : [0, 1] → R be defined as f,(x) = xn , for n e Z+, for x E [0, 1]. Define: f(x) := { 0, ге [0, 1) 1, x = 1. (i) Show that fn→f pointwise in [0,1]. (ii) Since continuity is preserved under uniform convergence, we can deduce that the sequenc
Suppose fn : [0, 1] → R be defined as f,(x) = xn , for n e Z+, for x E [0, 1]. Define: f(x) := { 0, ге [0, 1) 1, x = 1. (i) Show that fn→f pointwise in [0,1]. (ii) Since continuity is preserved under uniform convergence, we can deduce that the sequenc
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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![Suppose fn : [0, 1] → R be defined as f,(x)
Xn ,
for n e Z+, for x € [0, 1].
Define:
0,
x € [0, 1)
f(x) := {
1, x = 1.
(i) Show that f,n→f pointwise in [0,1].
(ii) Since continuity is preserved under uniform convergence, we can deduce that the
{fn}=1 does not uniformly converge to f. Use the ɛ- N definition of uniform convergence to
confirm that {fn} does not uniformly converge to f .
sequence](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9444f09-d451-4777-a726-2c28d1bfa98b%2Fee4ef8e9-fe5c-4c48-ba90-16d3c33ace14%2Fewov0g9_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose fn : [0, 1] → R be defined as f,(x)
Xn ,
for n e Z+, for x € [0, 1].
Define:
0,
x € [0, 1)
f(x) := {
1, x = 1.
(i) Show that f,n→f pointwise in [0,1].
(ii) Since continuity is preserved under uniform convergence, we can deduce that the
{fn}=1 does not uniformly converge to f. Use the ɛ- N definition of uniform convergence to
confirm that {fn} does not uniformly converge to f .
sequence
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