Math 2.0

statement theorem

Squeeze theorem

Statement

Let (an)(a_n), (bn)(b_n) and (cn)(c_n) be real sequences and let LL be a real number. Assume that there is a natural number NN such that an≤cn≤bna_n \le c_n \le b_n for all n≥Nn \ge N. Assume that an→La_n \to L and bn→Lb_n \to L as n→∞n \to \infty. Then cn→Lc_n \to L as n→∞n \to \infty.

Remarks

The theorem gives the existence of the limit of (cn)(c_n). It does not assume it. For example, −1/n≤sin⁡(n)/n≤1/n-1/n \le \sin(n)/n \le 1/n, so sin⁡(n)/n→0\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.

Explanations

L1 human

If a sequence is trapped between two sequences that go to the same limit, it must go there too. For large nn, the upper and lower sequences are both close to LL, and cnc_n lies between them.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatement
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  1. Sequence definition
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  4. Squeeze theorem statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Squeeze_theorem, Statement
  2. Web https://en.wikipedia.org/wiki/Limit_of_a_sequence, Properties

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