Limits are unique in Hausdorff spaces

Statement

Let $X$ be a Hausdorff space and let $(x_n)_{n \ge 1}$ be a sequence in $X$. Say that $(x_n)$ converges to $x \in X$ when for each neighbourhood $N$ of $x$ there is $n_0$ with $x_n \in N$ for all $n \ge n_0$. If $(x_n)$ converges to $x$ and to $y$, then $x = y$.

Remarks

Proof: if $x \neq y$, take disjoint open sets $U \ni x$ and $V \ni y$. For large $n$, the point $x_n$ is in both $U$ and $V$, which is impossible. The same holds for nets and filters. In fact, a space is Hausdorff if and only if each net in it has at most one limit. In an indiscrete space with at least two points, each sequence converges to each point.

Explanations

L1 human

A sequence cannot come close to two different points at the same time, if the points have disjoint neighbourhoods. So in Hausdorff spaces, you can speak of the limit of a sequence.