Exercise 512Topology › Metric and normed spacesL1Added 3 Nov 2026

Cauchy sequences

#Cauchy#sequences#metric

Statement

Define a Cauchy sequence and show that a convergent sequence is Cauchy.

Hint

For $\varepsilon>0$ , use $d(x_n,x)\to0$ and the triangle inequality.

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Definition. $(x_n)$ is Cauchy if $\forall\varepsilon>0,\ \exists N,\ \forall m,n\ge N,\ d(x_m,x_n)<\varepsilon$ .

Convergent $\Rightarrow$ Cauchy. If $x_n\to x$ , then for $m,n\ge N$ , $d(x_m,x_n)\le d(x_m,x)+d(x,x_n)<\varepsilon$ .

Conclusion. Every convergent sequence is Cauchy.

Note. The converse characterizes complete spaces.