Bounded iff Big-O of 1/Sequences

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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

$\blacksquare$