Archimedes's inequality

Let $m, n \in \mathbb{N}$. Prove that:

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

Solution

We note that:

\begin{align*}
\frac{n^{m+1}}{m+1} &=\int_{0}^{n}x^m \, {\rm d}x \\
 &= \sum_{k=1}^{n}\int_{k-1}^{k}x^m \, {\rm d}x\\
 &<\sum_{k=1}^{n}\int_{k-1}^{k}k^m \, {\rm d}x \\
 &=\sum_{k=1}^{n}k^m
\end{align*}

and similarly we have that:

 \begin{align*}
\frac{n^{m+1}}{m+1} &=\int_{0}^{n}x^m \, {\rm d}x \\
 &= \sum_{k=1}^{n}\int_{k-1}^{k}x^m \, {\rm d}x\\
 &>\sum_{k=1}^{n}\int_{k-1}^{k}(k-1)^m \, {\rm d}x \\
 &=\sum_{k=1}^{n}(k-1)^m
\end{align*}

ending the proof of the inequality.