Sum of Little-O Estimates

Theorem

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 little-$\oo$ notation.

Then:

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