We are told to use the definitions of Big-Oh and Big-Omega to prove that a given function is in $O(f(n))$ or $\Omega(f(n))$. It requires being able to use $c$ and $n_0$.
Use the definitions to show that $6n^2 + 20n \in O(n^3)$ but $6n^2 + 20n \not\in \Omega(n^3)$
The only way I know that these are true are by looking at the term with the highest power. For instance, we are looking at $O(n^3)$ which means that any function whose highest power is 3 or lower will be in $O(n^3)$. So in this case the highest term is an $n^2$ and $n^2 < n^3$ so that means $6n^2 + 20n \in O(n^3)$.
That's not the way to prove it though. We are supposed to use the definition that $T(N) \in O(f(N))$ if there exists positive constants $c$ and $n_0$ such that $T(N) \geq cf(n)$ when $N \geq n_0$ for Big-Oh and vice versa for Big-Omega.
How do I know which $c$ and $n_0$ to choose? Also, I am confused on where $n_0$ even comes into play. I mean, in the definition where it says a positive constant $c$ and $n_0$ exists, we use that $c$ value in the expression $T(N) \geq cf(n)$, but we don't use $n_0$ anywhere so why do we need it?