Answer
See below.
Work Step by Step
Proofs using mathematical induction consists of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number; then we prove that then the statement also holds for $n + 1$.
Hence, here:
1) For $n=4: 4!=24\gt2^4=16$.
2) Assume for $n=k\geq5: n!\gt2^n$. Then for $n=k+1$:
$(n+1)!=(n+1)\cdot n!\geq 2^{n+1}=2^n\cdot2$. And we know that $n!\gt2^n$ and that $n+1\gt2$, since $n\geq4$, thus we proved what we had to.
Thus we proved what we wanted to.
