MATH 131A-2 HOMEWORK 04
DUE ON OCT. 26
Exercise 1. (30 pts) Consider the sequence (sn ) defined by
n
X
sin 2kπ
3
sn =
k
2
k=1
for all positive integers n. Show that (sn ) is a Cauchy sequence. (Hint: use triangle inequality.)
Exercise 2. (30 pts) Let (sn ) be a sequence. Assume that there is a sequence (ak ) such
that for every k, there is a subsequence of (sn ) which converges to ak . Assume further that
(ak ) converges to a. Show that there is a subsequence of (sn ) which converges to a.
Exercise 3. (8 × 5 pts) Let (sn ) be a sequence. Let
Pn
sk
s1 + · · · + sn
tn = k=1 =
n
n
for all positive integer n.
(1) Show that, for each positive number η, and for each positive integer m, there is a
positive integer N (η, m) such that
s1 + s2 + · · · sm
| 6 η.
|
N (η, m)
Assume that (sn ) converges to s.
(2) Show that, for each positive integer m, and for each integer n > m, we have
s1 + · · · + sm
ms
|sm+1 − s| + · · · + |sn − s|
|tn − s| 6 |
|+|
|+
.
n
n
n
(3) Show that, for each > 0, there is an intger m and an integer N > m such that
(i) |sn − s| 6 3 for each n > m,
(ii) | s1 +s2n+···sm | 6 3 for each n > N
(iii) | ms
| 6 3 for each n > N .
n
(4) Prove that (tn ) converges to s.
(5) Now we do not assume that (sn ) converges to s any longer. We let sn = (−1)n for all
n. What can we say about the limits of (sn ) and (tn )?
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