---
tag: 0N1ZWFJDN5
kind: statement
role: proposition
name: Completeness via Cauchy sequences
msc: ["54E50"]
uses: [08EWT7J0SC, 0MCK8K2D7Z, 06JYPJ4WHH]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy_sequence
    locator: In a metric space
  - url: https://en.wikipedia.org/wiki/Complete_metric_space
    locator: Definition
---

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]].
Then $X$ is [[complete|08EWT7J0SC]] if and only if each [[Cauchy sequence|0MCK8K2D7Z]] in $X$ has a [[convergent|06JYPJ4WHH]] subsequence.

## Remarks

The key fact: if a Cauchy sequence $(x_n)$ has a subsequence that converges to $x$, then $x_n \to x$. For $\varepsilon > 0$, choose $N$ for $\varepsilon / 2$ in the Cauchy condition and a term $x_{n_k}$ with $n_k \ge N$ and $d(x_{n_k}, x) < \varepsilon / 2$.
A consequence: a metric space in which each sequence has a convergent subsequence is complete. Thus each [[sequentially compact|0CARW59WD2]] metric space is complete.

## Explanations

### L1 human

To show that a space is complete, you need not find the limit of a whole Cauchy sequence.
It is enough to find a limit of some subsequence, and then the whole sequence follows it.