Infinite Sequence in Countably Compact Space has Accumulation Point

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

Then $\left\langle{x_n}\right\rangle$ has an accumulation point in $T$.

Proof
Let $A \subseteq S$ be the range of $\left\langle{x_n}\right\rangle$:
 * $A = \left\{ {x_n: n \in \N}\right\}$

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

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

Hence, $y$ is an accumulation point of $\left\langle{x_n}\right\rangle$.

Otherwise, $A$ is countably infinite.

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

It follows that $\left\langle{x_n}\right\rangle$ has an accumulation point in $T$.