monotonically nonincreasing

A sequence $(s_n)$ is 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.

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

• $(s_n)=1,0,-1,-2,\mathrm{\dots }$ is monotonically nonincreasing. It is also monotonically decreasing.

• $(s_n)=1,1,1,1,\mathrm{\dots }$ is nonincreasing but not monotonically decreasing.

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

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

• $(s_n)=1,2,3,4,5,\mathrm{\dots }$ is not nonincreasing, rather it is nondecreasing (and monotonically increasing).