I'm confused if the following approach is mathematically correct
Suppose I have to prove $(\log_2n)! > n^a$, 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 $(\log_2n)! > n^a$, or should I extend the proof until I get a correct result in terms of $n$?