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

Question

I am stuck with the question below,

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

Answer 1

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.

Answer 2

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.