Get the right "upper" Sequence I want to find out the limit for the sequence $\frac{n!}{n^n}$ 
using the squeeze theorem.
My idea was:
$\frac{n}{n^n} \leq \frac{n!}{n^n} \leq \frac{n!+1}{n^n} $
So the limits of the smaller sequence and the bigger sequence are 0, thus
the limit of the original sequence is 0.
But is the bigger sequence $\frac{n!+1}{n^n}$ a good choice? Or is there
a more obvious sequence to choose? 

I think you are asking how large or small is $\frac{n!}{n^n}$, for large $n$.  
A rather clear answer is given by Stirling's formula that estimates $n!$.  The ratio of interest grows exponentially, at a rate of about $e^n$.
