Prove $2n-1 \leq n!$ is true using mathematical induction

Question

Prove the statement is true using mathematical induction: $$2n-1 \leq n!$$

My attempt: this is true for $n=1$.

Suppose it is true for $n$, i.e., $2n-1 \leq n!$

Now, $2n-1 \leq n!\implies 2n-1+2 \leq n!+2$

From here, how do I proceed?

Answer 1

$$\forall n\geq2, \ (n+1)!=n!(n+1)\geq(2n-1)(n+1)\geq2n+1$$

but for $n=2$ our statement is false, which says that $ \ \forall n\geq 3, \ n!\geq2n-1$.

Answer 2

for $n+1$ we get $2(n+1)-1=2n+1=2n-1+3 \le n!+3\le n!(n+1)$, which is true for $n \ge 2$