Limit Point of Sequence is Adherent Point of Range

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

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

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

Then $\alpha$ is an adherent point of $\left\{ {x_n: n \in \N}\right\}$.

Proof
By definition, $\left\{ {x_n: n \in \N}\right\}$ contains all the terms of $\left \langle {x_n} \right \rangle$.

In particular, $\left\{ {x_n: n \in \N}\right\}$ contains $\alpha$.

Thus by definition $\alpha$ is an adherent point of $\left\{ {x_n: n \in \N}\right\}$.