# Bounded iff Big-O of 1/Sequences

Jump to navigation
Jump to search

## 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 |

$\blacksquare$