Uniqueness of limits

Uniqueness of limits: Let $(X,d)$ be a metric space and let $(x_n)$ be a sequence in $X$. If $x_n\to x$ and $x_n\to y$, then $x=y$.

This lemma is a basic structural fact about metric convergence and is used everywhere (for example, to identify a limit by proving two different candidate limits).

Proof sketch: Assume $x\neq y$, so $d(x,y)>0$. Let $\varepsilon=d(x,y)/3$. For large $n$, we have $d(x_n,x)<\varepsilon$ and $d(x_n,y)<\varepsilon$. Then by the triangle inequality , $d(x,y)\le d(x,x_n)+d(x_n,y)<2\varepsilon=\frac{2}{3}d(x,y),$ a contradiction.