Prove inequalities $2\sqrt{n+1}-2 \leq 1 + 1/\sqrt{2} + 1/\sqrt{3} + .... + 1/\sqrt{n}\leq 2\sqrt{n}-1$ Prove that for any positive integer $n$,
$$
2\sqrt{n+1}-2 \leq 1 + 1/\sqrt{2} + 1/\sqrt{3} + .... + 1/\sqrt{n}\leq 2\sqrt{n}-1
$$
the middle part can be expressed as a sum. of $1/\sqrt{x}$ from 1 to $n$, which leads to a $Hn^{0.5}$. Im not sure how that helps.
I tried differentiating both end functions and found they are both increasing.. But it doesn't seem to help much... help pls

Hint: $2\sqrt{k+1} - 2\sqrt{k}<\dfrac{1}{\sqrt{k}} = \dfrac{2}{2\sqrt{k}} < \dfrac{2}{\sqrt{k}+\sqrt{k-1}} = 2\sqrt{k} - 2\sqrt{k-1}$. Now let $k$ runs from $1$ to $n$, and add up.
