Using the squeeze theorem to determine a limit $\lim_{n\to\infty} (n!)^{\frac{1}{n^2}}$
Hint:
Use the inequality
$$n^{\frac{n}{2}} \leqslant {n!} \leqslant {\left(\frac{n+1}{2}\right)^n}, \;\; n>1,$$
which can be proved by induction.
Hint:
$$\log n!=\sum_{k=2}^n\log k\le n\log n\implies e^{\frac1{n^2}\sum_{k=2}^n\log k}\le e^{\frac1n\log n}=\sqrt[n]{\log n}$$
But also
$$\sqrt[n]{\log n}\le\sqrt[n]n\xrightarrow[n\to\infty]{}1$$