Example 1

A sequence of points $(x_n)_{n \in \mathbf{N}}$ of a topological space is a mapping $n \to x_n$ of $\mathbf{N}$ into $X$. In analysis one frequently uses the notions of limit point and cluster point of such a mapping with respect to the Fréchet filter (§ 6, no. 1) on $\mathbf{N}$; if $y$ is a limit of $n \to x_n$ with respect to the Fréchet filter, $y$ is said to be a limit of the sequence $(x_n)$ as n tends to infinity, and we write $\lim_{n \to \infty} x_n = y$. A cluster point of the mapping $n \to x_n$ with respect to the Fréchet filter is called a cluster point of the sequence $(x_n)$.

Thus a point $y \in X$ is a limit (resp. cluster point) of a sequence $(x_n)$ of points of $X$ if it is a limit point (resp. cluster point) of the elementary filter associated with $(x_n)$ (§ 6, no. 8).

The point $y$ is a limit of a sequence $(x_n)$ in $X$ if and only if, for every neighbourhood $V$ of $y$ in $X$, all but a finite number of the terms of the sequence $(x_n)$ are in $V$, i.e. there is an integer $n_0$ such that $x_n \in V$ for all $n \geq n_0$. Likewise $y$ is a cluster point of the sequence $(x_n)$ if and only if, for every neighbourhood $V$ of $y$ in $X$ and every integer $n_0$, there is an integer $n \geq n_0$ such that $x_n \in V$.