Evaluate $$ \lim_{n\to\infty}\left(\dfrac{n}{n^2+1^2}+\dfrac{n}{n^2+2^2}+\dfrac{n}{n^2+3^2}+\cdots+\dfrac{n}{n^2+n^2}\right) $$
My Solution: Let $S$ denote the limit. Now observe
\begin{gather*} \frac{n}{n^2+n^2} < \frac{n}{n^2+1} \leq \frac{n}{n^2+1} \tag{1} \\ \frac{n}{n^2+n^2} < \frac{n}{n^2+2^2} < \frac{n}{n^2+1} \tag{2} \\ \vdots \\ \frac{n}{n^2+n^2} \leq \frac{n}{n^2+n^2} < \frac{n}{n^2+1} \tag{$n$} \end{gather*}
Now add all the equations and take the limit as $n\to\infty$. We obtain
$$ \lim_{n\to\infty}\frac{n^2}{n^2+n^2} < S < \lim_{n\to\infty}\dfrac{n^2}{n^2+1} $$
Hence
$$ \frac{1}{2} < S < 1 . $$
Question: Is my approach correct?
(I know that using Riemann Sum leads to the answer $S = \frac{\pi}{4}$.)
Also, can we solve it using Sandwich Theorem of Limit?