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

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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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
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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