In a Hausdorff Space, the Limit of a Sequence Is Unique
Theorem
A sequence $\left\{ x_{n} \right\}$ on a $T_{2}$-space $X$ does not converge to more than one point.
Explanation
Is there really any need to belabor how important the uniqueness of limits is? The very fact that a space possesses such a property is evidence that a Hausdorff space is useful.
What must be noted is that this differs slightly, in terms of phrasing, from ‘converges to exactly one point.’ If one wishes to use such an expression, one should rephrase it as ‘if the sequence converges, then it converges to exactly one point.’
Proof
Suppose that $\left\{ x_{n} \right\}$ converges to both of two distinct points $a,b \in X$.
Since $X$ is a $T_{2}$-space, $$ a \in U \\ b \in V \\ U \cap V = \emptyset $$ there exist open sets $U, V \subset X$ satisfying the above. Then there exist $n_{1} , n_{2} \in \mathbb{N}$ satisfying $$ n \ge n_{1} \implies x_{n} \in U \\ n \ge n_{2} \implies x_{n} \in V $$ However, since $$ n \ge \max \left\{ n_{1} , n_{2} \right\} \implies x_{n} \in U \cap V = \emptyset $$ this is a contradiction.
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