For the Fort space topology, let $\displaystyle (a_{n}) $be a sequence in X, such that the set of the sequence, $\displaystyle \{a_{n}:n\in\mathbb{N}\}$, is infinite.

Using the definition of convergence in topological space, prove that $\displaystyle (a_{n})$ has a subsequence which converges to p.

I'm struggling to make the transition from a sequence to a subsequence here.

I thought about writing $\displaystyle S=\{a_{n}:n\in\mathbb{N}\}$and $\displaystyle B=S\sim\{p\}$. B is infinite and $\displaystyle p\notin B$, so $\displaystyle \sim B $is a T-neighbourhood of p, and there is an $\displaystyle N\in\mathbb{N}$, such that, for all $\displaystyle n>N, ~a_{n}\in\sim B$.