Sum 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:
 * $a_n + c_n = \map \OO {\size {b_n} + \size {d_n} }$