Little-o notation

Question

Can anyone help me demonstrate these two statements?

$$ n! = o(n/2)^n $$

$$ n! = o(n/3)^n $$

I am sure about the first one but I don't know how to demonstrate it. As for the second one I am not sure whether it is true or not.

Answer 1

These statements have nothing to do with time complexity. They have everything to do with asymptotics.

Hint: Use Stirling's approximation: $$ n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n. $$ Use the fact that $2 < e < 3$.