I start with base saying that this is valid for one by simple algebraic calculus:
This is my base for $ n = 4$ $$ 2 \cdot 4 \leq 4! \Leftrightarrow 8 \leq 24 $$
In my hypotheses just say that exists a number $k$ that satisfies the expression for some $k\geq 4$: $$ 2k < k! $$
And my inductive step I multiply both sides from hypotheses by $(k+1)$ as follows: $$ (k+1) \cdot 2k < k! \cdot (k+1) $$ $$ \Leftrightarrow (k+1) \cdot 2k < (k+1)! $$
So by hypotheses we know that $k\geq 4$ soon can afirms that $2k(k+1) > 2k$. Given that $(k+1)! > 2k(k+1)$ and $2k(k+1) > 2k$ is proved by induction that $2n < n!$
I wrong anything or is insufficient to prove the enunciated?