Prove that $\left(\sum_{k=1}^n\,\left(\frac{k}{n}\right)^n\right)^{\frac{1}{n}}$ decreases as $n$ increases.

Question

Please help me to prove the following problem with inductive argument $$\left(\left(\frac{1}{n+1}\right)^{n+1}+\dots+\left(\frac{n+1}{n+1}\right)^{n+1}\right)^{\frac{1}{n+1}}<\left(\left(\frac{1}{n}\right)^n+\dots+\left(\frac{n}{n}\right)^n \right)^{\frac{1}{n}}\,.$$

I think we can use the following solution to prove the main problem:

$$\frac{1}{n+1}<\frac1n $$
$$(\frac1{n+1})^{n+1}<(\frac1n )^n$$

Thanks...