Exercise 514Topology › Metric and normed spacesL1Added 3 Nov 2026

Convergence in metric spaces

#convergence#sequences#metric

Statement

Define convergence of a sequence in a metric space and show that limits are unique.

Hint

Suppose $x_n\to x$ and $x_n\to y$ with $x\neq y$ ; take $\varepsilon=d(x,y)/2$ .

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Definition. $x_n\to x$ iff $d(x_n,x)\to0$ .

Uniqueness. If $x_n\to x$ and $x_n\to y$ , then $d(x,y)\le d(x,x_n)+d(x_n,y)\to0$ , so $d(x,y)=0$ , hence $x=y$ .

Conclusion. Limits in metric spaces are unique.