Limit Point of Sequence may only be Adherent Point of Range

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

Let $A \subseteq S$.

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 $\alpha$ may be only an adherent point of $A$ and not a limit point of $A$.

Proof
Let $T = \left({S, \tau}\right)$ be the discrete space on $S$.

Let $x \in S$.

Then by definition of discrete space:
 * $U = \left\{{x}\right\}$ is an open set of $T$.

Consider the sequence $\left\langle{x_n}\right\rangle$ defined as:
 * $\forall n \in \N: x_n = x$

That is:
 * $\left\langle{x_n}\right\rangle = \left({x, x, x, \ldots}\right)$

From Limit Point of Sequence in Discrete Space not always Limit Point of Open Set:
 * $x$ is not a limit point of $U$.

But from Limit Point of Sequence is Adherent Point of Range:
 * $x$ is an adherent point of $\left\{{x}\right\}$.

Hence the result.