Prove that for any positive integer $n$, $2\sqrt{n+1}-2\le 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\dots+\frac{1}{\sqrt{n}}\le 2\sqrt{n}-1$ Prove that for any positive integer n,$$ 2\sqrt{n+1} - 2 \le 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots  + \frac{1}{\sqrt{n}} \le 2\sqrt{n} - 1 $$
Could anyone give me a hint on this question? Does it have something to do with Riemann Sums?

Hints:
$$2(\sqrt{k+1}-\sqrt{k})=\frac{2}{\sqrt{k}+\sqrt{k+1}} < \frac{1}{\sqrt{k}}< \frac{2}{\sqrt{k-1}+\sqrt{k}}=2(\sqrt{k}-\sqrt{k-1})$$
