Limit of Sequence is Accumulation Point

Theorem
Let $$X$$ be a topological space.

Let $$A \subseteq X$$.

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

Let $$\alpha$$ be a limit point of $\left \langle {x_n} \right \rangle$.

Then $$\alpha$$ is also an accumulation point of $$\left \langle {x_n} \right \rangle$$.

Proof
Let $$\alpha$$ be a limit point of $\left \langle {x_n} \right \rangle$

Then by definition of limit, $$\left \langle {x_n} \right \rangle$$ converges to $$\alpha$$.

By definition of converge that means:
 * for any open set $$U \subseteq T$$ such that $$\alpha \in U$$: $$\exists N \in \R: n > N \implies x_n \in U$$.

As there is an infinite number of values of $$n > N$$, there are an infinite number of terms of $$\left \langle {x_n} \right \rangle$$.

Hence the result, from definition of accumulation point.