,

$$\begin{eqnarray*} \sqrt{1+\frac{1}{n^{2}}+\frac{1}{\left(n+1\right)^{2}}} & = & \sqrt{\frac{n^{4}+2n^{3}+3n^{2}+2n+1}{n^{2}\left(n+1\right)^{2}}}\\ & = & \sqrt{\frac{\left(n^{2}+n+1\right)^{2}}{n^{2}\left(n+1\right)^{2}}}\\ & = & \frac{n^{2}+n+1}{n\left(n+1\right)}\\ & = & 1+\left(\frac{1}{n}-\frac{1}{n+1}\right) \end{eqnarray*}$$

,

$$\begin{eqnarray*} S_{n} & = & \sqrt{1+\frac{1}{1^{2}}+\frac{1}{2^{2}}}+\sqrt{1+\frac{1}{2^{2}}+\frac{1}{3^{2}}}+\cdots+\sqrt{1+\frac{1}{n^{2}}+\frac{1}{\left(n+1\right)^{2}}}\\ & = & \left(1+\frac{1}{1}-\frac{1}{2}\right)+\left(1+\frac{1}{2}-\frac{1}{3}\right)+\left(1+\frac{1}{3}-\frac{1}{4}\right)+\cdots+\left(1+\frac{1}{n}-\frac{1}{n+1}\right)\\ & = & n+1-\frac{1}{n+1} \end{eqnarray*}$$

,

$$\begin{eqnarray*} \lim_{n\rightarrow\infty}\frac{1}{n}\left(S_{n}\right) & = & \lim_{n\rightarrow\infty}\frac{1}{n}\left(n+1-\frac{1}{n+1}\right)\\ & = & \lim_{n\rightarrow\infty}\left(1+\frac{1}{n}-\frac{1}{n\left(n+1\right)}\right)\\ & = & \lim_{n\rightarrow\infty}1+\lim_{n\rightarrow\infty}\frac{1}{n}-\lim_{n\rightarrow\infty}\frac{1}{n\left(n+1\right)}\\ & = & 1+0+0 \end{eqnarray*}$$