Definition:Big-O Notation/Implied Constant

Definition
Let $f$ and $g$ be real functions such that $f$ is big-$\OO$ of $g$.

From the definition of the limit of a function, it can be seen that this is also equivalent to:
 * $\exists c \in \R: c > 0, k \ge 0: \forall n > k: \map f n \le c \map g n$

For some fixed $k$ (appropriate to the function under consideration) the infimum of such $c$ is called the implied constant.