Big-O Notation for Sequences Coincides with General Definition

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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.


The following are equivalent:

$(1): \quad$ $a_n = O(b_n)$, 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


Proof