Proving inequailty by induction

Question

Prove by induction that $$ (n!)^2 \geqslant n^n$$ for all positive natural numbers.

Answer 1

Note that $$(n+1)^{n+1}/n^n=(n+1)\cdot\left(\frac{n+1}n\right)^n$$$$=(n+1)\cdot\left(1+\frac1n\right)^n<(n+1)\cdot e$$

And, $$(n+1!)^2/(n!)^2=(n+1)^2$$

So, $(n+1)^2>(n+1)\cdot e$ when $n+1>e$, which happens when $n>1$. Can you take it from here?