Limit of a Sequence

Definition: Limit of a Sequence

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

A point $l\in X$ is a limit of $(x_n{)}_{n\in I}$ iff $(x_n{)}_{n\in I}$ converges to $l$.

Warning: Non-Uniqueness of Limits

In general, a sequence may have multiple different limits.

NOTATION

$$\underset{n\to \infty }{\lim }{x}_n=l{x}_n\to l$$