monotonically nonincreasingMonotonicallyNonincreasing2002-02-18 21:35:042005-08-16 00:55:35Definition\usepackage{amssymb}
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%\usepackage{xypic}A sequence $(s_n)$ is \emph{monotonically nonincreasing} if
$$ s_m \le s_n \text{ for all } m > n $$
Similarly, a real function $f(x)$ is monotonically nonincreasing if
$$ f(x) \le f(y) \text{ for all } x > y $$
Compare this to monotonically decreasing.
\textbf{Conflict note.} In some contexts, such as \cite{NIST}, this is called \emph{monotonically decreasing} (in turn, our ``monotonically decreasing'' is called ``strictly decreasing''). This is unfortunately counter-intuitive, since a sequence or function that is ``flat'' (such as $f(x) = 1$) is somehow ``decreasing.'' Beware!
\section{Examples}
\begin{itemize}
\item $(s_n) = 1, 0, -1, -2, \ldots$ is monotonically nonincreasing. It is also monotonically decreasing.
\item $(s_n) = 1, 1, 1, 1, \ldots$ is nonincreasing but {\bf not} monotonically decreasing.
\item $(s_n) = (\frac{1}{n+1})$ is nonincreasing (note that $n$ is nonnegative).
\item $(s_n) = 1, 1, 2, 1, 1, \ldots$ is {\bf not} nonincreasing. It also happens to fail to be monotonically nondecreasing.
\item $(s_n) = 1, 2, 3, 4, 5, \ldots$ is {\bf not} nonincreasing, rather it is nondecreasing (and monotonically increasing).
\end{itemize}
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\bibitem{NIST} ``\PMlinkexternal{monotonically decreasing}{http://www.nist.gov/dads/HTML/monotoncdecr.html},'' from the NIST Dictionary of Algorithms and Data Structures, Paul E. Black, ed.
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