Give an argument for ∫n0xpdx≤1+2p+3p+⋯+np≤∫n+10xpdx\int_{0}^{n} x^p dx \leq 1 +2^{p} + 3^{p} + \cdots+ n^{p}\leq \int_{0}^{n+1} x^p dx

For any $n$ and $p\geq 0$ give an argument that the following is true:

I’m having trouble even beginning this question. My first thought it to somehow meld this with the squeeze theorem, but, again, am not sure how to begin and show any real work. Any insight is very much appreciated.

Now, since $p>0$, $x^p$ is an increasing function for $x>0$. Thus $m^p<(m+1)^p$ for all $m>0$. Then, we have