The Squeeze Rule

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 a_n, b_n and c_n are sequences such that a_n\leq b_n \leq c_n for all n. If a_n\to l and c_n \to l, then b_n \to l.

Proof.

Let x_n=b_n-a_n and y_n=c_n-a_n. Then \displaystyle y_n = c_n-a_n \to  l - l =0 .
So as y_n \to 0 , given any \varepsilon >0 there exists N such that |y_n-0| <\varepsilon for all n\geq N. But for n\geq N and as 0\leq x_n \leq y_n,

|x_n-0|=|x_n|\leq |y_n| = |y_n-0|<\varepsilon .

Therefore, x_n\to 0 and so \displaystyle b_n = x_n+a_n \to 0+ l =l. \hfil \square .

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