Limit Point of Sequence is Adherent Point of Range

Theorem

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

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

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

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

Proof

By definition of Limit of sequence:

$\forall U \in \tau : \exists N \in \N : \forall n \ge N : x_n \in U$

Hence:

$\forall U \in \tau : U \cap \set{x_n: n \in \N} \ne \O$.

By definition $\alpha$ is an adherent point of $\set{x_n: n \in \N}$.

$\blacksquare$