How to prove $n < n!$ if $n > 2$ by induction?

I am stuck with the question below,

Prove by mathematical induction that $n<n!$ for $n>2$.

First, for $n=3$ you have $3< 3!=6 $. Suppose that for some $k$ it holds that $k<k!$ then $$ (k+1)! = (k+1)k!>(k+1)k\geq k+1 $$ since $k\geq 3$. Could you please tell which step is unclear to you in this proof? By elaborating on it maybe we can learn how to use induction.

If this is homework and the professor specifically said to use induction, then disregard this answer, I suppose. Otherwise, the statement can be proven directly without induction.

Given any $n \geq 3$, we can write $n! = n(n-1)!$ and be confident that $n-1 \geq 2$ (so we aren't making inappropriate use of $0!$). From this expression, it is clear that $n! > n$, since $n!$ is equal to $n$ times some number strictly greater than 1.