It's a very classic problem for students learning induction to prove the following statement:
"Prove $n! > 2^n$ for all $n \geq 4$"
I went on to prove a related, still quite simple example:
"Prove $n! > n^2 2^n$ for all $n \geq 8$."
This isn't very different than the first proof, but a little bit more involved, I suppose.
My question is: Is there a general way to prove a statement such as:
"Prove $n! > P(n) x^n$ for all $n \geq \alpha$," where $P(n)$ is a general polynomial in terms of $n, x \geq 1$ and $\alpha$ is some threshold for the base case to be true. In the first example, $\alpha = 4$, and in the second example, $\alpha = 8$.
How might we prove such a statement to be true by induction? There seems to be lots of variables going on, which may make the proof more complicated.