Show that $\{ s_{n}\} =\left\{(-1) ^{n}\dfrac{1}{n}\right\}$ converges and find its limit.

Solution We seek two sequences that "squeeze" $\{ s_{n}\} = \left\{ (-1) ^{n}\dfrac{1}{n}\right\}$ as $n$ becomes large. We begin with $\vert s_{n}\vert$: $$\begin{eqnarray*} \vert s_{n}\vert &=&\left\vert (-1) ^{n}\dfrac{1}{n}\right\vert \leq \dfrac{1}{n} \\[5pt] -\dfrac{1}{n} &\leq &(-1) ^{n}\dfrac{1}{n}\leq \dfrac{1}{n} \end{eqnarray*}$$

Notice that $s_{n}$ is bounded by $\{a_{n}\} =\left\{ -\dfrac{1}{n}\right\}$ and $\{b_{n}\} =\left\{ \dfrac{1}{n}\right\}$. Since $a_{n}\leq s_{n}\leq b_{n}$ for all $n$ and $\lim\limits_{n\rightarrow \infty }a_{n}=\lim\limits_{n\rightarrow \infty }\left( -\dfrac{1}{n}\right) =0$, and $\lim\limits_{n\rightarrow \infty }b_{n}=\lim\limits_{n\rightarrow \infty }\dfrac{1}{n}=0$, then by the Squeeze Theorem, the sequence $\{ s_{n}\} =\left\{ (-1) ^{n}\dfrac{1}{n}\right\}$ converges and $\lim\limits_{n\rightarrow \infty } s_n=0$.