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 $1^2 + 2^2 + ..... + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + ...... + n^2$.

$f(x)$ is an increasing function, so left Riemann sums under-approximate, and right Riemann sums over-approximate. Using a step size of 1, the inequality is
$$\sum_{i=1}^n \sqrt{i-1} < \int_0^n \sqrt{x} \, \mathrm{d}x < \sum_{i=1}^{n} \sqrt{i}.$$
