# monotonically nonincreasing

A sequence $(s_{n})$ is monotonically nonincreasing if

 $s_{m}\leq s_{n}\text{ for all }m>n$

Similarly, a real function $f(x)$ is monotonically nonincreasing if

 $f(x)\leq f(y)\text{ for all }x>y$

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 counter-intuitive, since a sequence or function that is “flat” (such as $f(x)=1$) is somehow “decreasing.” Beware!

## 1 Examples

• $(s_{n})=1,0,-1,-2,\ldots$ is monotonically nonincreasing. It is also monotonically decreasing.

• $(s_{n})=1,1,1,1,\ldots$ is nonincreasing but not monotonically decreasing.

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

• $(s_{n})=1,1,2,1,1,\ldots$ is not nonincreasing. It also happens to fail to be monotonically nondecreasing.

• $(s_{n})=1,2,3,4,5,\ldots$ 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 MonotonicallyNonincreasing 2013-03-22 12:22:32 2013-03-22 12:22:32 akrowne (2) akrowne (2) 9 akrowne (2) Definition msc 40-00 monotone nonincreasing nonincreasing MonotonicallyNondecreasing