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Lebl-Basic-Analysis-I
· Problem 7.3.11
Metric Spaces
Mathematics · sequences-and-convergence
Problem
Suppose
{
V
n
}
n
=
1
∞
\{V_n\}_{n=1}^\infty
{
V
n
}
n
=
1
∞
is a sequence of open sets in
(
X
,
d
)
(X, d)
(
X
,
d
)
such that
V
n
+
1
⊃
V
n
V_{n+1} \supset V_n
V
n
+
1
⊃
V
n
for all
n
n
n
. Let
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
be a sequence such that
x
n
∈
V
n
+
1
∖
V
n
x_n \in V_{n+1} \setminus V_n
x
n
∈
V
n
+
1
∖
V
n
and suppose
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
converges to
p
∈
X
p \in X
p
∈
X
. Show that
p
∈
∂
V
p \in \partial V
p
∈
∂
V
where
V
=
⋃
n
=
1
∞
V
n
V = \bigcup_{n=1}^\infty V_n
V
=
⋃
n
=
1
∞
V
n
.
Related theory
Sequences Convergence Cauchy Sequences Subsequences
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