Convergence of a sequence in a metric space

Let $(X,d)$ be a metric space . A sequence $(x_n)$ in $X$ converges to a point $a\in X$ if

We write $\lim_{n\to\infty}x_n=a$ or $x_n\to a$.

Convergence in metric spaces is the foundation for defining closure via sequences and for studying Cauchy sequences .

Examples:

• In $\mathbb{R}$, $x_n=1/n$ converges to $0$.
• In any metric space, a constant sequence $x_n=x$ converges to $x$.
• In the discrete metric, $x_n\to a$ iff $x_n=a$ for all sufficiently large $n$.