The squeeze rule is one of the most useful tools in the study of sequences. It allows us to show that a sequence converges to a limit if we ‘sandwich’ it between two other sequences that converge to the same limit. In effect the sequence gets squeezed between the other two.

### Theorem: The Squeeze Rule

Suppose that and are sequences such that for all . If and , then .

### Proof.

Let and . Then

So as , given any there exists such that for all . But for and as ,

Therefore, and so