I am trying to find the constants $n_0$ and $c$ to show that some given functions belong to the $O(\cdot)$ equivalent class. But, while it seems easy, I am not sure whether I am allowed to do what I will showcase below, or rather, what decides which constants I should take into consideration. For example:
$2^{100}n \in O(n^2)$.
The definition is: $\exists c, \exists n_0 / n>n_0,f(n) \leq c g(n)$.
In my example:
If I take $n_0=1$ then $c \geq 2^{100} $.
If I take $n_0=10$ then $c \geq \frac {2^{100}}{10}$.
So, if I am asked to find the constants $n_0$ and $c$, with nothing more asked (no extra conditions etc), how can I decide which values to consider?