Prove that the Fibonacci function $F(n)$ is NOT $\Omega(n)$

Question

Is there a straightforward proof that follows directly from the definition of the Fibonacci sequence (i.e., $F(0)=1, F(1)= 1, F(n) = F(n-1)+F(n-2) \text{ for } n \geq 2$), without using, say, the closed-form expression?

Answer 1

It's clear that the Fibonacci numbers are increasing, so $$ F(n+2) > 2F(n). $$

That's good enough to guarantee exponential growth.