---
tag: 0QY87R9CY3
kind: statement
role: proposition
name: Convergent sequences are Cauchy
msc: [40A05]
uses: [0AZ3R52593, 0QZMTN5AXX]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy_sequence
    locator: Other properties
---

## Statement

Let $(a_n)$ be a [[convergent sequence|0AZ3R52593]] of real numbers.
Then $(a_n)$ is a [[Cauchy sequence|0QZMTN5AXX]].

## Remarks

Let $L$ be the limit and let $\varepsilon > 0$. Take $N$ with $|a_n - L| < \varepsilon/2$ for $n \ge N$. Then $|a_m - a_n| \le |a_m - L| + |L - a_n| < \varepsilon$ for $m, n \ge N$.
The statement holds in each metric space.
The converse holds in $\mathbb{R}$, but it fails in $\mathbb{Q}$ and in other spaces that are not complete.

## Explanations

### L1 human

If all late terms are close to one number $L$, then they are also close to each other.
Two terms that are each within $\varepsilon/2$ of $L$ are within $\varepsilon$ of each other.