Convergence of Sequences

Convergence

Definition: Convergence of Sequences

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

We say that $(x_n{)}_{n\in I}$ is convergent if there exists some $L\in X$ such that for each neighborhood $N(L)$ there exists some $N_0\in I$ with

$$x_n\in N(L)\forall n\ge N_0.$$

Definition: Limit

Any such $L$ is called a limit of $(x_n{)}_{n\in I}$. We also say that $(x_n{)}_{n\in I}$ converges to $L$.

NOTATION

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

Warning: Non-Uniqueness of Limits

In general, $(x_n{)}_{n\in I}$ may have multiple different limits.