Skip to Main Content
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 37 #37
Sequences and Series 3-37
The above discussions show that all the possible limits of partial
sums of the above rearrangement must lie in [α, β] only. Hence
lim inf
n→∞
t
n
= α and lim sup
n→∞
t
n
= β.
This completes the proof of our theorem.
SOLVED EXERCISES
1. Suppose {x
n
}is a real sequence satisfying
|
x
n+1
x
n
|
α
|
x
n
x
n1
|
for
some fixed α (0, 1). Show that {x
n
} converges.
Solution: Since a real sequence converges if and only if it is Cauchy,
it is enough to show that {x
n
} is a Cauchy sequence, i.e. we show that
|
x
m
x
n
|
0asm, n →∞.
Let m = n + p and p 0 be integers.
From the given hypothesis,-3pc]Please ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Make: Calculus

Make: Calculus

Joan Horvath, Rich Cameron
Complex Analysis

Complex Analysis

ITL Education

Publisher Resources

ISBN: 9781299447561Publisher Website