Bounded iff Big-O of 1/Sequences

Theorem

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

The following statements are equivalent:

$(1): \quad a_n$ is bounded

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

Proof

\(\ds a_n\)

\(\text {is}\)

\(\ds \text {bounded}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists k \in \R: \, \)

\(\ds \size {a_n}\)

\(\le\)

\(\ds k\)

Definition of Bounded Sequence

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists k \in \R: \, \)

\(\ds \size {a_n}\)

\(\le\)

\(\ds k \cdot \size 1\)

\(\ds \leadstoandfrom \ \ \)

\(\ds a_n\)

\(=\)

\(\ds \map \OO 1\)

Definition of Big-$\OO$ Notation

$\blacksquare$