Showing $\left\{ \frac{1}{n^n}\sum_{k=1}^{n}{k^k} \right\}_{n \in \mathbb{N}}\rightarrow 1$

I would like to prove:

$$\left\{ \frac{1}{n^n}\sum_{k=1}^{n}{k^k} \right\}_{n \in \mathbb{N}}\rightarrow 1$$

I found a proof applying Stolz criterion but I need to use the fact that:

$$\left\{\left(\frac{n}{n+1}\right)^n\right\}\text{ is bounded}\tag{\ast}$$

I would like to calculate this limit without using $(\ast)$ and preferably using basic limit criterion and properties.

(Sorry about mispellings or mistakes, English is not my native language.)

Obviously $$\frac{1}{n^n}\sum_{k=1}^{n}k^k \geq 1,$$ while: $$\frac{1}{n^n}\sum_{k=1}^{n}k^k\leq \frac{1}{n^n}\sum_{k=1}^{n}n^k\leq\frac{1}{n^n}\cdot\frac{n^n}{1-\frac{1}{n}}=\frac{n}{n-1}.$$
$$n^n\le\sum_{k=1}^nk^k\le n^n+(n-1)^{n-1}+(n-2)(n-2)^{n-2}$$