Sum of Big-O Estimates/Sequences

Theorem
Let $(a_n),(b_n),(c_n),(d_n)$ be sequences of real or complex numbers.

Let:
 * $a_n = O(b_n)$
 * $c_n = O(d_n)$

where $O$ denotes big-O notation.

Then $a_n + c_n = O(|b_n| + |d_n|)$.