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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_03” 2011/5/22 22:50 page 14 #14
3-14 Real Analysis
Theorem 3.3.6 (viii) lim sup
n→∞
s
n
lim sup
n→∞
t
n
. This proves (a). Similarly
we can prove (b).
Theorem 3.3.14 If {s
n
}
n=1
and {t
n
}
n=1
are sequences of extended
real numbers, then
lim inf
n→∞
s
n
+ lim inf
n→∞
t
n
lim inf
n→∞
(s
n
+ t
n
) lim sup
n→∞
(s
n
+ t
n
)
lim sup
n→∞
s
n
+ lim sup
n→∞
t
n
Proof Since the second inequality is trivial, we shall prove the first
and the last inequalities. Let M
n
= inf
jn
s
j
, P
n
= inf
jn
t
j
, V
n
= inf
jn
(s
j
+t
j
)
and α = sup M
n
= lim inf
n→∞
s
n
, β = sup P
n
= lim inf
n→∞
t
n
, γ = sup V
n
=
lim inf
n→∞
(s
n
+ t
n
). Now our claim is that α + β γ . It is clear that
M
n
+ P
n
V
n
γ for all n. Since our sequences are from extended
real numbers we have several cases to consider. However, in order ...
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Publisher Resources

ISBN: 9781299447561Publisher Website