Bounded iff Big-O of 1/Sequences

From ProofWiki
Jump to navigation Jump to search


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


\(\ds a_n\) \(\text {is}\) \(\ds \text {bounded}\)
\(\ds \leadstoandfrom \ \ \) \(\ds \exists k \in \R: \, \) \(\ds \size {a_n}\) \(\le\) \(\ds k\) 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