Convergence of a sequence in a metric space

Definition

Let $(X, d)$ be a metric space, let $(x_n)_{n \ge 1}$ be a sequence of points of $X$, and let $x \in X$. The sequence $(x_n)$ converges to $x$ when for each real $\varepsilon > 0$ there is an integer $N$ such that $d(x_n, x) < \varepsilon$ for all $n \ge N$. Then $x$ is a limit of $(x_n)$, with the notation $x_n \to x$ or $\lim_{n \to \infty} x_n = x$.

Remarks

Equivalently, $x_n \to x$ if and only if $d(x_n, x) \to 0$ in $\mathbb{R}$. A sequence in a metric space has at most one limit. This follows from the triangle inequality, and also from the fact that limits are unique in Hausdorff spaces. For $X = \mathbb{R}$ with the usual metric, this is the usual limit of a real sequence.

Explanations

L1 human

A sequence converges to $x$ when its terms come as close to $x$ as you want and stay close. For each tolerance $\varepsilon$, only finitely many terms are farther than $\varepsilon$ from $x$.