Ruminating on Constants in Big-O Notation

When trying to understand the definition of the big-O notation, it is natural to ask questions regarding the constants hiddenhidden in the big-O notation.

Constants in big-O notation not unique

There are many legitimate values of $n'$ for which $f(n)$ is $O(g(n))$, in the example shown below. We can use the slider to see some valid and invalid values of $n'$ (we show only integer values).

In fact, in general, given that $f(n)$ is $O(g(n))$, there are infinitely many choices for the threshold value and the constant $c'$.

If something is true for all values of $n \geq 5$, then it must be true for all values of $n \geq 6$, all values of $n \geq 7$, and so on. Similarly, there are infinitely many choices of $c'$. If $f(n) \leq 3 g(n)$ ...