Product of Big-O Estimates/Sequences

Theorem
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:
 * $a_n c_n = \map \OO {b_n d_n}$