<s>[INST] Find $\lim_{n\to \infty}({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}})$ strong textFind $\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)$.
I do know it is bounded by $1$. I tried using the sandwich rule with no success. How can I solve it?
 [/INST] Note 
$$\lim_\limits{n\to \infty}\frac{n}{\sqrt{n^2+n}}\le\lim_\limits{n\to \infty}\left({1\over \sqrt{n^2+1}}+{1\over \sqrt{n^2+2}}+\cdots+{1\over \sqrt{n^2+n}}\right)\le\lim_\limits{n\to \infty}\frac{n}{\sqrt{n^2+1}}$$
Since 
$$\lim_\limits{n\to \infty}\frac{n}{\sqrt{n^2+n}}=\lim_\limits{n\to \infty}\frac{1}{\sqrt{1+\frac{1}{n}}}=1$$
and
$$\lim_\limits{n\to \infty}\frac{n}{\sqrt{n^2+1}}=\lim_\limits{n\to \infty}\frac{1}{\sqrt{1+\frac{1}{n^2}}}=1$$
we have that the limit of the original is $1$ by the sandwich rule.
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