\begin{definition}[Definition:O Notation/Big-O Notation/Real/Infinity]
Let $f$ and $g$ be real-valued or complex-valued functions defined on a neighborhood of $+ \infty$ in $\R$.
The statement:
:$\map f x = \map \OO {\map g x}$ as $x \to \infty$
is equivalent to:
:$\exists c \in \R_{\ge 0}: \exists x_0 \in \R: \forall x \in \R: \paren {x \ge x_0 \implies \size {\map f x} \le c \cdot \size {\map g x} }$
That is:
:$\size {\map f x} \le c \cdot \size {\map g x}$
for $x$ sufficiently large.
This statement is voiced '''$f$ is big-O of $g$''' or simply '''$f$ is big-O $g$'''.
\end{definition}