Limit of a sequence

Let $(X,d)$ be a metric space and let $(x_n)$ be a sequence in $X$. A point $x\in X$ is the limit of $(x_n)$ if

One writes $x=\lim_{n\to\infty}x_n$ or $x_n\to x$.

The limit (when it exists) summarizes the eventual behavior of a sequence. In metric spaces, a sequence has at most one limit (see convergent sequence ).

Examples:

• $\lim_{n\to\infty} (1/n)=0$ in $\mathbb{R}$.
• If $x_n=2$ for all $n$, then $\lim_{n\to\infty}x_n=2$.
• The sequence $(-1)^n$ has no limit in $\mathbb{R}$.