# Countably Compact First-Countable Space is Sequentially Compact/Proof 2

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## Theorem

A countably compact first-countable topological space is also sequentially compact.

## Proof

Let $T = \struct {S, \tau}$ be a countably compact first-countable topological space.

By definition of sequentially compact, it is sufficient to show that every infinite sequence in $S$ has a convergent subsequence.

Let $\sequence {s_n}$ be any sequence in $S$.

By Infinite Sequence in Countably Compact Space has Accumulation Point, $\sequence {s_n}$ has an accumulation point $p \in S$.

As $T$ is first-countable, $p$ has a countable local basis, say:

- $\set {V_n: V_1 \supseteq V_2 \supseteq V_3 \supseteq \cdots}$

Then a subsequence $\sequence {s_{n_i} }$, where $s_{n_i} \in V_i$, converges to $p$.

$\blacksquare$

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 3$: Countability Axioms and Separability