CM221A
ANALYSIS I
Exercise Sheet 1
1. Prove that, for every
δ >
0, there exists
n
∈
N
such that
n

1
< δ
.
2. Let
a
1
= 1 and
a
n
+1
=
a
n
a
n
+ 2
for all
n
≥
2. Prove that the sequence is
decreasing and ﬁnd its limit.
3. Making references to any theorems about convergence which you use, evaluate
lim
n
→∞
2
n
3
+
n
2
+
n

n
5
n
3
+ 2
n

1 + 2
n
5
and
lim
n
→∞
2
n
+ 3
3
n
+ 2
.
4. Let
k
∈
R
and
b >
1. Prove that lim
n
→∞
n
k
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 Spring '09
 Topology, convergent sequence, limn→∞ nk b−n

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