monotonically nonincreasing
A sequence $({s}_{n})$ is monotonically nonincreasing if
$${s}_{m}\le {s}_{n}\text{for all}mn$$ 
Similarly, a real function $f(x)$ is monotonically nonincreasing if
$$f(x)\le f(y)\text{for all}xy$$ 
Compare this to monotonically decreasing.
Conflict note. In some contexts, such as [1], this is called monotonically decreasing (in turn, our “monotonically decreasing” is called “strictly decreasing”). This is unfortunately counterintuitive, since a sequence or function^{} that is “flat” (such as $f(x)=1$) is somehow “decreasing.” Beware!
1 Examples

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$({s}_{n})=1,0,1,2,\mathrm{\dots}$ is monotonically nonincreasing. It is also monotonically decreasing.

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$({s}_{n})=1,1,1,1,\mathrm{\dots}$ is nonincreasing but not monotonically decreasing.

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$({s}_{n})=(\frac{1}{n+1})$ is nonincreasing (note that $n$ is nonnegative).

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$({s}_{n})=1,1,2,1,1,\mathrm{\dots}$ is not nonincreasing. It also happens to fail to be monotonically nondecreasing.

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$({s}_{n})=1,2,3,4,5,\mathrm{\dots}$ is not nonincreasing, rather it is nondecreasing (and monotonically increasing).
References
 1 “http://www.nist.gov/dads/HTML/monotoncdecr.htmlmonotonically decreasing,” from the NIST Dictionary of Algorithms and Data Structures, Paul E. Black, ed.
Title  monotonically nonincreasing 

Canonical name  MonotonicallyNonincreasing 
Date of creation  20130322 12:22:32 
Last modified on  20130322 12:22:32 
Owner  akrowne (2) 
Last modified by  akrowne (2) 
Numerical id  9 
Author  akrowne (2) 
Entry type  Definition 
Classification  msc 4000 
Synonym  monotone nonincreasing 
Synonym  nonincreasing 
Related topic  MonotonicallyNondecreasing 