Using the Squeeze Theorem for Sequences

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 \(\lim\limits_{n\rightarrow \infty } s_n=0 \).