<s>[INST] A simple double inequality with roots Prove or disprove for integer $n$ greater than $1$: 
$$\sqrt{1} + \sqrt{2} + ...+ \sqrt{n-1} < \frac{2n\sqrt{n}}{3} < \sqrt{1} + \sqrt{2} + ...+ \sqrt{n} $$
I think I know a solution, but I am looking for different approaches.
EDIT: Inspired by Prove that $1^2 + 2^2 + ..... + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + ...... + n^2$.
 [/INST] by induction : 
for LHS: if $n=k ,\sum _{i=1}^k \sqrt{i-1} \le \dfrac{2k\sqrt{k}}{3}$
when $n=k+1$, we have to prove $\dfrac{2k\sqrt{k}}{3}+\sqrt{k} \le \dfrac{2(k+1)\sqrt{k+1}}{3} \iff 3k\ge 4$ which is true when $k\ge2$
similar method to RHS.
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