I was wondering how to prove that $$n! \geq n^{\frac{n}{2}}\quad\forall n \geq 1$$ without analytic methods that relies on asympotic comparison or use of logarithms/exponentials.
Trying by induction I was stuck immediately after the induction hypothesis because I was unable to give an estimate of the following :
$$(n+1)! = n!(n+1) \geq n^{\frac{n}{2}}(n+1)$$
Is there anything I'm missing ? Any help of tip would be appreciated, as well as other methods that don't rely on seeing the inequality as an analysis task,
I'm seeking for ''arithmetic'' or ''algebraic'' demonstrations.