
“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
n−1
|
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 ...