Proof by induction: $n! > n^2$ for $n\ge 4$ Proof by induction: $n! > n^2$ for $n\ge 4$
Basis step:
if $n=4$
$4\cdot3\cdot2\cdot1 > 4^2$
$24 > 16$
I don't know how to do the inductive step.

Inductive step:
$$(n+1)! = (n+1)n! > (n+1)n^2 $$
But clearly, $n^2 > n + 1$ for integer $n \ge 2$. You could prove this rigorously by showing that the curve $n^2 - n - 1$ lies strictly above the $x$ axis for $n \ge 2$.
Hence,
$$(n+1)!  > (n+1)^2$$
