I'm trying to prove an inequality that I believe to be the case. The inequality is as follows

$$k! < p^{k-1}\prod_{j = 1}^{k-1}(p^{n-1}-j)$$

where $1 < k \leq p^{n-1}$, $p$ a fixed odd prime and $n > 1$ a fixed positive integer.

I've shown that it's true for a couple of small values for $p$ and $n$ but I'm not sure how I might tackle the problem in more generality.

If there's a counter-example that I've missed, I'd be overjoyed if anyone could point it out. Hints towards a solution would be preferable!


Lemma. $k< 3^{k-1}\le p^{k-1}$ for $k>1$
Proof. By mathematical induction.

Base step is satisfied because $2<3^{2-1}=3$, and inductive step is done because $3^n=3(3^{n-1})>3(n-1)>n$. Q.E.D.

Because of lemma, it suffices to prove that $$(k-1)!\le\prod_{j = 1}^{k-1}(p^{n-1}-j)$$ and this is equivalent with $1 \le \binom{p^{n-1}-1}{k-1}$, and this is easily satisfied as $k-1 \le p^{n-1}-1$.


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