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I'm confused if the following approach is mathematically correct

Suppose I have to prove $(\log n)! > n^a$, where $a$ is a constant

I can assume $n = 2^k$ which leads to $k! > c^k$, where $c = 2^a$

The conclusion is true since factorials have a higher rate of growth than exponentials.

My doubt is, is it correct to terminate the proof there and conclude $(\log n)! > n^a$, or should I extend the proof until I get a correct result in terms of $n$?

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    $\begingroup$ That would be fine, if you have proved beforehand that $k!$ grows faster than $c^k$ for any $c$. $\endgroup$
    – vonbrand
    Commented Feb 18, 2020 at 22:53

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