Uniqueness of limits

Proposition. A sequence in a metric space has at most one limit.

More precisely: if $(x_n)$ converges to $a$ and also converges to $b$, then $a=b$.

Proof sketch. Fix $\varepsilon>0$. For large $n$, both $d(x_n,a)<\varepsilon/2$ and $d(x_n,b)<\varepsilon/2$. Then

Since this holds for all $\varepsilon>0$, the lemma nonnegative below every epsilon gives $d(a,b)=0$, hence $a=b$.