My proof of "Is $2^{2n} = O(2^n)$?"

Question

I was wondering if my proof for this problem was correct.

Let $x = 2^n$. Then the problem reduces to is $x^2 = O(x)$, which is clearly false.

Answer 1

Note that $g(n) = O(2^n)$ if $\lim_{n \to \infty} \frac{g}{2^n} = c $ for some constant c. If you take the limit of the function you have you get infinity, which implies that your function $h(n) = \omega (2^n)$, which is a stronger condition.