Let $f(n) = 2n^4 − 4n^3 + 16n^2 − 64^n + 3$.
(a) Using the constant $c = 1.9$, prove that $f(n) \in \Omega(n^4)$.
Just by looking at it it is clear to me that this is true as for $f(n) \in \Omega(n^4)$ to be true, $f(n) \ge 1.9n^4$ which is true. I just don't know how to prove it on paper. Any help would be greatly appreciated!
(b) Let $\epsilon$ be any positive real number. Prove that $f(n) \in \Omega(n^4)$ using the constant $c = 2 − \epsilon$ in the definition of big-Omega. Hint: Your $n_0$ is allowed to depend on $\epsilon$.
(a) I would use the limit test. By showing $\lim_{n \to \infty} \frac{f(n)}{1.9n^{4}} \to \infty$, you get $f(n) \in \Omega(n^{1.4})$. Alternatively, you can prove this by induction.
(b) Again, the limit test is your friend.