Convergent sequence

A convergent sequence $(x_n)$ in a topological space $X$ converges to a point $x\in X$ if for every neighborhood $U$ of $x$, there exists $N$ such that $x_n\in U$ for all $n\ge N$.

In a metric space $(X,d)$, this is equivalent to $d(x_n,x)\to 0$. In a Hausdorff space , limits of convergent sequences are unique (see uniqueness of limits ).

Examples:

• In $\mathbb{R}$ with the usual metric, the sequence $x_n=1/n$ converges to $0$.
• In a space with the discrete metric, a sequence converges if and only if it is eventually constant.