How to understand the use of constant $c$ in the equation of $f(n) \leqslant cg(n)$ in Asymptotic notation?
Solution 1:
For example, we want to say that $f(n) = 2n^2+1$ is $O(n^2)$.
This is true under the current definition because for all $n \ge 1$, $f(n) \le 3n^2$. Here, $n_0=1$ and $c=3$.
There's other choices of $n_0$ and $c$ that would work. For example, for all $n \ge 10$, $f(n) \le 2.01 n^2$. Here, $n_0 = 10$ and $c=2.01$.
However, there's not going to be any $n_0$ such that for all $n \ge n_0$, $2n^2+1 \le n^2$. So we need the $c$.