The squeeze lemma

Limits part 1: Infinite sequences: Techniques

The squeeze lemma

Watch this animation of the sequence $b_n = \frac{1+(-1)^n}{n}$ .

The sequence consists of two parts: the constant sequence $0$ and the sequence $\frac{2}{n}$ . These two sequences both converge to $0$ as $n\to\infty$ , so we would like to conclude that $\lim_{n\to\infty} b_n = 0$ as well. The squeeze lemma justifies this:

Suppose that $(a_n)_{n\in\mathbb N}, (b_n)_{n\in\mathbb N}$ and $(c_n)_{n\in\mathbb N}$ are three sequences with $a_n \leq b_n \leq c_n$ for all $n$ . If

$\lim_{n\to\infty} a_n = \lim_{n\to\infty} c_n,$ then $b_n$ converges as well and the limit equals the limit of the other two sequences:

$\lim_{n\to\infty} a_n =\lim_{n\to\infty} b_n= \lim_{n\to\infty} c_n\text.$

In our example of $b_n = \frac{1+(-1)^n}{n}$ we choose $a_n = 0$ and $c_n = \frac{2}{n}$ since

$0 \leq \frac{1+(-1)^n}{n} \leq \frac{2}{n}$ for all $n$ . The lemma now implies that

$0 \leq \lim_{n\to\infty} b_n \leq 0$ So the limit equals $0$ .