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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_10” 2011/5/22 23:26 page 12 #12
10-12 Real Analysis
Theorem 10.2.13 Let
n=0
a
n
x
n
and
n=0
b
n
x
n
converge in
S ={x/|x| , δ>0}. Let E ={x S/
n=0
a
n
x
n
=
n=0
b
n
x
n
}.
If E has a limit point in S, then a
n
= b
n
for all n and hence E = S.
Proof Let f (x) =
n=0
c
n
x
n
, where c
n
= a
n
b
n
so that f (x) = 0 for
x E. Let A be the set of all limit points of E in S and B = S \ A.
Since A is closed (note that A is the derived set of E), B is open. If
only we can prove that A is open, then S = A B gives a separation
for the connected interval S. This shows that either A =∅or B =∅.
However by hypothesis A =∅and hence B =∅or that S = A. We now
observe that the continuity of f on S (note that f is even differentiable
in S) implies that S = A E (note ...
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Publisher Resources

ISBN: 9781299447561Publisher Website