Definition:Big-O Notation/Sequence

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

The statement:


 * $a_n = \mathcal O \left({b_n}\right)$

is equivalent to:
 * $\exists c \in \R: c \ge 0 : \exists n_0 \in \N : \left({n \ge n_0 \implies \left\vert{a_n}\right\vert \le c \cdot \left\vert{b_n}\right\vert}\right)$

That is:
 * $\left\vert{a_n}\right\vert \le c \cdot \left\vert{b_n}\right\vert$

for all sufficiently large $n$.

Also defined as
Some authors require that $b_n$ be nonzero for $n$ sufficiently large.

Remark
It can be shown that this definition coincides with the general definition if $\N$ is given the discrete topology.