Infinite Sequence in Countably Compact Space has Accumulation Point

Corollary to Countably Infinite Set in Countably Compact Space has Omega-Accumulation Point
Let $T = \struct {S, \tau}$ be a countably compact topological space. Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $S$.

Then $\sequence {x_n}$ has an accumulation point in $T$.

Proof
Let $A \subseteq S$ be the range of $\sequence {x_n}$:
 * $A = \set {x_n: n \in \N}$

If $A$ is finite, then consider the equality:
 * $\displaystyle \N = \bigcup_{y \mathop \in A} \set {n \in \N: x_n = y}$

Therefore, there exists a $y \in A$ such that $\set {n \in \N: x_n = y}$ is an infinite set.

Hence, $y$ is an accumulation point of $\sequence {x_n}$.

Otherwise, $A$ is countably infinite.

Then $A$ has an $\omega$-accumulation point in $T$.

It follows that $\sequence {x_n}$ has an accumulation point in $T$.