Big-O Notation for Sequences Coincides with General Definition

Theorem
Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences of real or complex numbers.

Let $\N$ be given the discrete topology.


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