Sequences, series, summability Proposition

Epsilon-N definition of the limit of a sequence

Statement

Let $(a_n)$ be a real sequence and let $L$ be a real number. Then $(a_n)$ converges to $L$ , in the sense of the limit of a sequence, if and only if the following holds. For each real number $\varepsilon > 0$ there is a natural number $N$ such that $|a_n - L| < \varepsilon$ for all $n \ge N$ .

Remarks

The proof uses two facts. Each interval $(L - \varepsilon, L + \varepsilon)$ is open and contains $L$ . Each open interval that contains $L$ contains such an interval. In a metric space $(X, d)$ the same statement holds with $d(a_n, L)$ in place of $|a_n - L|$ . The number $N$ depends on $\varepsilon$ . A smaller $\varepsilon$ in general needs a larger $N$ .

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Sequence definition
Real sequence definition
Limit of a sequence definition
Epsilon-N definition of the limit of a sequence statement

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Tag	`09G6KHS6GY`
URI	`m2://09G6KHS6GY`
Kind	statement, proposition
Writing	W0 stub
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 09G6KHS6GY`.
Uses	Limit of a sequence

Epsilon-N definition of the limit of a sequence

Statement

Remarks

Reading path

Where this leads

Further reading

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