Sum of Little-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 little-$\oo$ notation.

Then:
 * $a_n + c_n = \map \oo {\size {b_n} + \size {d_n} }$

Proof
Let $\epsilon > 0$.

Then by definition of little-$\oo$ notation:
 * $\exists n_1 \in \N: \paren {n \ge n_1 \implies \size {a_n} \le \epsilon \cdot \size {b_n}}$
 * $\exists n_2 \in \N: \paren {n \ge n_2 \implies \size {c_n} \le \epsilon \cdot \size {d_n}}$

For $n \ge \max \set {n_1, n_2}$:

Hence by definition of little-$\oo$ notation:
 * $a_n + c_n = \map \oo {\size {b_n} + \size {d_n} }$