Definition (Convergence)

Let $(X,\mathscr{T})$ be a topological space.

1. A sequence $\{ x_{j} \}_{j\in \mathbb{N}}$ converges to $x\in X$ in the Topology $\mathscr{T}$ if, for each neighbourhood $\mathcal{U}$ of $x$, there exists $N\in \mathbb{N}$ such that $x_{j}\in \mathcal{U}$ for each $j\ge N$. If $\{ x_{j} \}_{j\in \mathbb{N}}$ converges to $x$, we may write $$\lim_{ j \to \infty } x_{j}=x.$$
2. A sequence $\{ x_{j} \}_{j\in \mathbb{N}}$ is convergent if it converges to some point in $X$.