Limit Point of Sequence may only be Adherent Point of Range

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$.

Let $\sequence {x_n}$ be a sequence in $A$.

Let $\alpha$ be a limit point of $\sequence {x_n}$.

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

Proof
Let $T = \struct {S, \tau}$ be the discrete space on $S$.

Let $x \in S$.

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

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

That is:
 * $\sequence {x_n} = \tuple {x, x, x, \ldots}$

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 $\set x$.

Hence the result.