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

Proof
Since:


 * $a_n = \map \OO {b_n}$

there exists a positive real number $C_1$ and natural number $N_1$ such that:


 * $\size {a_n} \le C_1 \size {b_n}$

for all $n \ge N_1$.

Similarly, since:


 * $c_n = \map \OO {d_n}$

there exists a positive real number $C_2$ and natural number $N_2$ such that:


 * $\size {c_n} \le C_2 \size {d_n}$

for all $n \ge N_2$.

Let:


 * $N = \max \set {N_1, N_2}$

Then, for $n \ge N$ we have:

Let:


 * $C = \max \set {C_1, C_2}$

Then:

So:


 * $\size {a_n + c_n} \le C \size {\size {b_n} + \size {d_n} }$

for $n \ge N$.

So:


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