Big-O Notation for Sequences Coincides with General Definition

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

Let $\N$ be given the discrete topology.

The following statements are equivalent:

$(1): \quad a_n = \map \OO {b_n}$, 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