1. Write the definition of
infimum
by modeling it on the definition of
supremum
. State Lemma 1.3.7 for Infimums.
2. Let
s
= inf(
A
) and define

A
=
{
x
:
x
∈
A
}
. Prove that sup(

A
) =

s
.
3. Prove that inf
(
{
1 +
1
n
2
:
n
∈
N
}
)
= 1 using Lemma 1.3.7.
4. Find the limit of the sequence (
√
2
,
1 +
√
2
,
1 +
1 +
√
2
, . . .
) [Hint:
this sequence can be defined recursively.]
5. Find examples of the following, if possible:
(a) a sequence which is Cauchy, but not monotone.
(b) a sequence which is monotone, but not Cauchy.
(c) a sequence which is bounded, but not Cauchy.
(d) two sequences (
x
n
) and (
y
n
) such that
x
n
diverges,
y
n
converges,
and (
x
n
+
y
n
) converges
(e) two sequences (
x
n
) and (
y
n
) which both converge, but (
x
n
y
n
)
diverges.
(f) two sequences (
x
n
) and (
y
n
) which both diverge, but (
x
n
y
n
) con
verges.
(g) A monotone sequence which diverges but has a convergent sub
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 Winter '08
 Givens
 Mathematical analysis, Cauchy sequence, Cauchy

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