Convergence of Sequences

Definition: Convergence of Sequences

Let $(X,\tau )$ be a topological space and let $(x_n{)}_{n\in \mathbb{N}}$ be a sequence of points in $X$.

We say that $(x_n{)}_{n\in \mathbb{N}}$ is convergent if there exists some $x\in X$ iff for each neighbourhood $N(x)$ there exists some $N_0\in \mathbb{N}$ such that $x_n\in N(x)$ for all $n\ge N_0$.

We also say that $(x_n{)}_{n\in \mathbb{N}}$ converges to $x$.