# Product of Big-O Estimates

## 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:

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