Let $x\_n \colon= \sqrt[n]{n!} $
for all $n \in \mathbb{N}$. 

Then how to determine rigorously whether the sequence $\left( x\_n \right)\_{n \in \mathbb{N} }$ converges or diverges? 

And, how to find $\lim\_{n \to \infty} x\_n$, rigorously?

By *rigorously* I mean using the same machinery as has been developed by Rudin until Chap. 3, where he discusses sequences. 

My Attempt: 

> 
> We note that, for all $n \in \mathbb{N}$ such that $n > 1$, we have
>  $$ 1 \leq \left( x\_n \right)^n \leq n^{n-1}, $$
>  and so 
>  $$ 1 \leq x\_n \leq n^{ (n-1)/n } = \frac{ n }{\sqrt[n]{n} }. $$
>  However, although 
>  $$ \lim\_{n \to \infty} \sqrt[n]{n} = 1, $$
>  we also have 
>  $$ \lim\_{n \to \infty} n = +\infty. $$
>  Thus the squeeze theorem is not applicable. 
> 
> 
> 

Or, can we find some majorizing sequence converging to $1$?

Is this sequence monotonic?