# Product of Big-O Estimates

## Theorem

### Sequences

Let $\sequence {a_n}, \sequence {b_n}, \sequence {c_n}, \sequence {d_n}$ be sequences of real or complex numbers.

Let:

$a_n = \map \OO {b_n}$
$c_n = \map \OO {d_n}$

where $\OO$ denotes big-O notation.

Then:

$a_n c_n = \map \OO {b_n d_n}$

### Real Analysis

Let $c$ be a real number.

Let $f, g : \hointr c \infty \to \R$ be real functions.

Let $R_1 : \hointr c \infty \to \R$ be a real function such that $f = \map \OO {R_1}$.

Let $R_2 : \hointr c \infty \to \R$ be a real function such that $g = \map \OO {R_2}$.

Then:

$f g = \map \OO {R_1 R_2}$