<s>[INST] mathematical induction proof of a square vs factorial So lets say I have $$ n^{2} \le n! $$
For what positive integers is this not true? $n=2$ and $ 3$
Base case? $$n=4 \implies  16 \le 24 $$
What is the inductive hypothesis and how do I show the inductive proof? Thanks
 [/INST] For, $ x_0 = 4 $, $ n^2 \le n! $ is true.
Now, given, $n^2 \le n!$, 
$(n+1)^2 = n^2 + 2n + 1$, 
and, $ n^2 \le n! \implies n^2(n+1)^2 \le (n+1)^2n!$
$\implies (n+1)^2 \le (\frac{1}{n^2} + \frac{1}{n}) * (n+1)!$
$n > 4 => \frac{1}{n^2} < \frac{1}{16} $ and $\frac{1}{n} < \frac{1}{4} $, So,
$ (\frac{1}{n^2} + \frac{1}{n}) \lt \frac5{16} < 1$
As, $ (n+1)^2 \le \frac5{16} * (n+1)! \implies (n+1)^2 \le (n+1)!$
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