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Analysis 2
Miscellaneous Problems 4
1. Let
U
be a free (non-principal) ultrafilter on
Z
; define
μ
:
P
(
Z
)
→ {
0
,
1
}
by
μ
(
A
)
=
1 when
A
∈ U
and
μ
(
A
)
=
0 when
A
<
U
. Prove that
μ
is
σ
-additive i
ff
U
is closed under countable intersections.
2. Let (
a
n
)
n
∈
N
be a bounded real sequence. Prove that the following conditions are
equivalent:
(i) there exists a subsequence (
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- Spring '09
- LARSON
- Topology, lim, Supremum, Limit of a sequence, subsequence, real sequence
-
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