Big O Proof , f(n) = 2n + 1 and I have to prove f(n) is O n^2
It all depends on the case and how easy it is to search $n_0$ or use $c$, in your case you can show that $2n + 1$ belongs to $n^2$, taking the constant $c = 1$, so that in the proof of BigO, $g (n)$ dominates $f(n)$, what is the same $2n + 1 \le c (n^2)\to  2n + 1 \le n^2$; now you can search a $n_0$.