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Lebl-Basic-Analysis-I
· Problem 7.3.13
Metric Spaces
Mathematics · sequences-and-convergence
Problem
Let
(
X
,
d
)
(X, d)
(
X
,
d
)
be a metric space and
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
a sequence in
X
X
X
. Prove that
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
converges to
p
∈
X
p \in X
p
∈
X
if and only if every subsequence of
{
x
n
}
n
=
1
∞
\{x_n\}_{n=1}^\infty
{
x
n
}
n
=
1
∞
has a subsequence that converges to
p
p
p
.
Related theory
Sequences Convergence Cauchy Sequences Subsequences
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