How to determine the $\lim_{n\to \infty} \frac{1+2^2+\ldots+n^n}{n^n}=1$. I stuck to do this,

$$\lim_{n\to \infty} \frac{1+2^2+\ldots+n^n}{n^n}=1.$$

The only thing I have observed is $$ 1\le \lim_{n\to \infty} \frac{1+2^2+\ldots+n^n}{n^n}$$ I am unable to get its upper estimate so that I can apply Sandwich's lemma. 

$$1+2^2+3^3+\cdots(n-2)^{n-2}+(n-1)^{n-1}+n^n<\\n^{n-2}+n^{n-2}+n^{n-2}+\cdots n^{n-2}+n^{n-1}+n^n.$$
When divided by $n^n$, these terms are bounded by $\dfrac n{n^2}+\dfrac1n+1$.
