Definition:Convergent Sequence/Topology

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

Let $\left \langle {x_n} \right \rangle_{n \in \N}$ be an infinite sequence in $X$.

Then $\left \langle {x_n} \right \rangle$ converges to the limit $\alpha \in X$ iff:
 * $\forall U \in \tau: \alpha \in U \implies \left({\exists N \in \R: \forall n \in \N: n > N \implies x_n \in U}\right)$

This can be alternatively stated:

$\left \langle {x_n} \right \rangle$ converges to the limit $\alpha \in X$ iff:
 * $\forall U \in \tau: \alpha \in U \implies \left\{{n \in \N: x_n \notin U}\right\}$ is finite

Such a sequence is convergent.