Asymptotics - Big Omega

Question

I have a question about Asymptotics involving big Omega... How do I need to approach this equation in order to prove it?

$$n \cdotΩ(f(n)) = Ω(n\cdot f(n))$$

Thank you very much for your answers!

Answer 1

$g(n) = \Omega ( f(n))$ means that $\dfrac{g(n)}{f(n)}$ is bounded in a suitable sense. Formally you have also that $\dfrac{\Omega(f(n))}{f(n)}$ is bounded. What you've written just states that $$ \frac{n \Omega(f(n))}{n f(n)} = \frac{\Omega (f(n))}{f(n)}$$ is bounded.

Answer 2

Using the definition of Big-Omega:

By definition, $f(n) \in \Omega(f(n))$, so $f(n) \leq 1 * f(n)$, for all $n$. Now multiply both sides by $n$, to get $n f(n) \leq 1 * n f(n)$. Again, we set our constant $C = 1$ and the inequality holds for all $n$. In fact, it's really equality. So $n \Omega(f(n)) = \Omega(n f(n))$.