# Bounded iff Big-O of 1/Sequences

## Theorem

Let $\sequence {a_n}$ be a sequence of real or complex numbers.

The following are equivalent:

$(1): \quad a_n$ is bounded
$(2): \quad a_n = \map \OO 1$, where $\OO$ denotes big-$\OO$ notation

## Proof

 $\displaystyle a_n$ $\text {is}$ $\displaystyle \text {bounded}$ $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \exists k \in \R: \,$ $\displaystyle \size {a_n}$ $\le$ $\displaystyle k$ Definition of Bounded Sequence $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \exists k \in \R: \,$ $\displaystyle \size {a_n}$ $\le$ $\displaystyle k \cdot \size 1$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle a_n$ $=$ $\displaystyle \map \OO 1$ Definition of Big-$\OO$ Notation

$\blacksquare$