$2\sqrt n-2<1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+....+\frac{1}{\sqrt n}<2\sqrt n-1$ If $n\in N$,then prove that $2\sqrt n-2<1+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}+....+\frac{1}{\sqrt n}<2\sqrt n-1$
How should i prove this inequality,neither AM-HM nor any other inequality theorem is working here.

Denote the sum by $S_n$. Looking at the graph of $f(x)=\frac{1}{\sqrt{x}}$ it is clear that you have $$S_n <1+\int_1^{n} \frac{1}{\sqrt{x}} dx=1+[2\sqrt{x}]_1^n=2\sqrt{n}-1$$
Can you also derive the lower bound by a slight variation of this method?
