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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_09” 2011/5/22 23:21 page8—#8
9-8 Real Analysis
Theorem 9.3.9 Let {f
n
} be a sequence of functions defined on a
common set E
R and that lim
n→∞
f
n
(x) = f (x)(pointwise). Put
M
n
= sup
xE
|f
n
(x) f (x)|.
Then f
n
f uniformly on E as n →∞if and only if M
n
0 as
n →∞.
Proof Let f
n
f uniformly on E as n →∞. Given >0 there exists
a stage N such that
|f
n
(x) f (x)| <
2
x E and n N .
It follows that M
n
= sup
xE
|f
n
(x) f (x)|≤
2
<for all n N . This
shows that M
n
0asn →∞.
Conversely, if M
n
0asn →∞then given >0 there exists a
stage N such that M
n
<n N . It follows that |f
n
(x) f (x)| <
for all x E and for all n N or that f
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Publisher Resources

ISBN: 9781299447561Publisher Website