Definition:Big-O Notation/Sequence/Definition 2

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

$a_n$ is big-$\OO$ of $b_n$ :
 * $\exists c \in \R_{\ge 0}: \exists n_0 \in \N: \forall n \in \N: \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$

That is:
 * $\size {a_n} \le c \cdot \size {b_n}$

for all sufficiently large $n$.

This is denoted:
 * $a_n \in \map \OO {b_n}$

Also see

 * Equivalence of Definitions of Big-O Notation for Sequences