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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 page9—#9
Sequences and Series 3-9
Case 2: α =−∞. In this case a
n
→−∞as n →∞and hence
given any real number M we have a stage N such that a
n
< M for
n N . Thus α
n
M for n N . (Note that {α
n
} is a decreasing
sequence). Thus α
n
→−∞as n →∞. However, by Theorem 3.3.7,
η = inf
n1
α
n
= lim
n→∞
α
n
=−∞=α.
Case 3: −∞ <α<. We merely prove that η satisfies both the
properties of α as described in (vi) so that η = α. Let γ>ηbe given.
Now γ is not a lower bound for {α
n
/n 1}. Thus we can get N N
such that α
N
. Since {α
n
} is decreasing, we have α
n
α
N
for n N . Hence the sequence {α
n
} is eventually in (−∞, γ). Now
let δ<ηbe given. Since η α
n
for n 1, we also ave δ<α
n
for
each n 1. From the definition of α
n
is not
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Publisher Resources

ISBN: 9781299447561Publisher Website