Sequences, series, summability Proposition

Convergent sequences are Cauchy

Builds on Convergent sequence Cauchy sequence

Statement

Let $(a_n)$ be a convergent sequence of real numbers. Then $(a_n)$ is a Cauchy sequence.

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.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Sequence definition
Real sequence definition
Cauchy sequence definition
Limit of a sequence definition
Convergent sequence definition
Convergent sequences are Cauchy statement

Tag	`0QY87R9CY3`
URI	`m2://0QY87R9CY3`
Kind	statement, proposition
Writing	W0 stub
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 0QY87R9CY3`.
Uses	Convergent sequence, Cauchy sequence

Convergent sequences are Cauchy

Statement

Remarks

Reading path

Further reading

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