I am looking for a nice proof of this inequality: \begin{equation} \sum_{k=1}^{n-1} k^n < n^n, \quad n > 0. \end{equation}
Example: $1^4 + 2^4 + 3^4 = 1+16+81 = 98 < 4^4 = 256.$
I am looking for a nice proof of this inequality: \begin{equation} \sum_{k=1}^{n-1} k^n < n^n, \quad n > 0. \end{equation}
Example: $1^4 + 2^4 + 3^4 = 1+16+81 = 98 < 4^4 = 256.$