Odd-Even Topology is Weakly Countably Compact

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $S$.

Then $T$ is weakly countably compact.

Proof
Let $A \subseteq S$ such that $A$ is infinite.

Let $x \in A$.

By definition, the odd-even topology is a partition topology.

So $U$ is a union of sets of the form $\left\{{2k-1, 2k}\right\}$.

Now if $x \in U$, it will be of the form $2k-1$ or $2k$.

So there will exist $y \in U$ of the form $2k$ or $2k-1$.

So, by definition, $x$ is a limit point of $A$.

So, by definition, $T$ is weakly countably compact.