Exponents and Factorials: Mathematical Induction Proof

Easy one here...

Prove that $2^{n}< n!$ for all $n\geq 4$, $n\in \mathbb{N}$.

Step 1: Show P(4) is true.
Our base case is k = 4. $2^{4}<4!$ $\Rightarrow 16 < 24$. OK.

Step 2: Show that (if we assume P(k) is true), P(k+1) is true. (Induction hypothesis). This means we should have $2^{k+1}<(k+1)!$.

$$2^{k+1}<(k+1)!$$
can be rewritten as
$$2\cdot2^{k}<(k+1) k!$$

For $k\geq 4$, 2 < k + 1, and by the induction hypothesis, we assume $2^{k}< k!$, so we can see that
$$2^{n}< n!$$
for all $n\geq 4$, $n\in \mathbb{N}$

QED