Explanation on a solution from inducting a statement?

Question

In this post, I am dealing with the same exact question, expect of Integers it is Nat (the solution still holds)

Prove that $n!>n^2$ for all integers $n \geq 4$.

However, the top-rated answer is hard to understand in part A from the (expand) step, I don't get at all how he converted $k^2$ to (k+1) at all. Even the steps after it are beyond me.

Answer 1

hint

Since for $ n\ge 4$, we have

$$n!= n.(n-1)!$$

You just need prove that

$$(\forall n\ge 4)\;\; (n-1)! > n$$

or, in an equivalent way

$$(\forall n\ge 3)\;\; n!> n+1$$

which is much easier.