Sequences, series, summability Definition

Cauchy sequence

Definition

Let $(a_n)$ be a real sequence. The sequence is a Cauchy sequence if the following holds for each real number $\varepsilon > 0$ . There is a natural number $N$ such that $|a_m - a_n| < \varepsilon$ for all natural numbers $m, n \ge N$ .

Remarks

In a metric space $(X, d)$ , the definition is the same with $d(a_m, a_n)$ in place of $|a_m - a_n|$ . The definition does not mention a limit. In $\mathbb{R}$ , the Cauchy criterion shows that the Cauchy sequences are exactly the convergent sequences. It is not enough that $|a_{n+1} - a_n| \to 0$ . The sequence $a_n = \sqrt{n}$ has this property, but it is not Cauchy.

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Sequence definition
Real sequence definition
Cauchy sequence definition

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Tag	`0QZMTN5AXX`
URI	`m2://0QZMTN5AXX`
Kind	definition
Writing	W1 stated
MSC	40A05
Defines	Cauchy sequence
Uses	Real sequence

Cauchy sequence

Definition

Remarks

Reading path

Where this leads

Further reading

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