So I need to proof the following inequality

$\displaystyle \sum_{k\ =\ 0}^{n-1} k^{p} < \frac{n^{p+1}}{p+1} < \sum_{k\ =\ 0}^n k^{p}$

If i start it by induction, I get to a point where I have to proove that

$\displaystyle k^p (k+p+1) < (k+1)^{p+1} $ (with the first part)

And I have no idea where to go from here. Any suggestions?