true or false : is $2^n = Ω(3^n) $

Question

$2^n = Ω(3^n) $

let $n= 0$

then

$1<1*c$ for any $c>1$

so my answer is yes, but the textbook answer implies I am wrong. may I know where I am doing it wrongly?

Answer 1

$\dfrac{2^n}{3^n}=\left(\dfrac{2}{3}\right)^n$.

Suppose this were were greater than some positive $k$ for an infinite number of positive integers $n$. Then taking logs, $n \lt -\dfrac{\log(k)}{\log(3/2)}$ for an infinite number of positive integers $n$. But that is false, as only a finite number of positive integers can be less than any given number.

So $2^n \not = \Omega(3^n)$