Big-O Notation for Sequences Coincides with General Definition

Theorem
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers.

Let $\N$ be given the discrete topology.


 * $(1): \quad a_n = \map \OO {b_n}$, where $\OO$ denotes big-$\OO$ notation for sequences
 * $(2): \quad a_n = \map \OO {b_n}$, where $\OO$ stands for the general definition of big-$\OO$ notation