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