Limit of Subsequence equals Limit of Sequence

Theorem
Let $T = \left({A, \vartheta}\right)$ be a Hausdorff space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $T$.

Let $l \in A$ such that $\displaystyle \lim_{n \to \infty} x_n = l$.

Let $\left \langle {x_{n_r}} \right \rangle$ be a subsequence of $\left \langle {x_n} \right \rangle$.

Then $\displaystyle \lim_{r \to \infty} x_{n_r} = l$.

That is, the limit of a convergent sequence in a Hausdorff space equals the limit of any subsequence of it.

Remark
Even if $T$ is not a Hausdorff space, subsequences of sequences converge to the same points. However, this limit may not be unique. The proof is identical.