Sequences, series, summability Sequences and limits Theorem

Squeeze theorem

Statement

Let $(a_n)$ , $(b_n)$ and $(c_n)$ be real sequences and let $L$ be a real number. Assume that there is a natural number $N$ such that $a_n \le c_n \le b_n$ for all $n \ge N$ . Assume that $a_n \to L$ and $b_n \to L$ as $n \to \infty$ . Then $c_n \to L$ as $n \to \infty$ .

Remarks

The theorem gives the existence of the limit of $(c_n)$ . It does not assume it. For example, $-1/n \le \sin(n)/n \le 1/n$ , so $\sin(n)/n \to 0$ . The same statement holds for the limit of a function at a point. Other names are the sandwich theorem and, in some countries, the two policemen theorem.

Reading path

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Sequence definition
Real sequence definition
Limit of a sequence definition
Squeeze theorem statement

Where this leads

114 pages build on this one.

Tag	`0XMF00T4XQ`
URI	`m2://0XMF00T4XQ`
Aliases	`sandwich-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	40A05
Status	This copy of the site has no build data. Run `m2 status 0XMF00T4XQ`.
Uses	Limit of a sequence
Same as	Wikidata Q1065257 Wikipedia Squeeze theorem

Squeeze theorem

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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