Uniqueness of limits

Statement

Let $(a_n)$ be a real sequence and let $L$ and $L'$ be real numbers. If $(a_n)$ converges to $L$ and $(a_n)$ converges to $L'$, then $L = L'$.

Remarks

The proof takes $\varepsilon = |L - L'|/2$ in the epsilon-N definition. If $L \ne L'$, then for large $n$ the term $a_n$ would be closer than $\varepsilon$ to both numbers, which is impossible. The statement holds in each metric space and, more generally, in each Hausdorff topological space. The statement fails in some spaces that are not Hausdorff. In a set with the trivial topology, each sequence converges to each point.

Explanations

L1 human

A sequence cannot approach two different numbers at the same time. If the two numbers are a distance $d$ apart, the terms cannot be within $d/2$ of both of them.