# Why non-increasing is decreasing?

As far as I know, by definition, non-decreasing means increasing and non-increasing means decreasing. My general question is: why some people use non-increasing and non-decreasing?

In fact, it raises some confusing to me. For example, the sequence

$$1,2,3,4$$

is increasing and thus the sequence

$$4,1,2,3$$

is non-increasing. So, based on the definition, it is decreasing, but it is not.

• What you think of as nonincreasing isn't what I think of as non-increasing. Commented Dec 19, 2018 at 15:21
• "non-increasing means decreasing" NO : $0,0,0,\ldots$ is non-increasing but not decreasing. Commented Dec 19, 2018 at 15:22
• "nonincreasing" $\neq$ "not increasing" Commented Dec 19, 2018 at 15:24

This is a bit of confusing terminology.

"Non-increasing," unfortunately, does not mean "not increasing" - it means "never increasing." So, for example, the sequence $$3,3,2,1$$ is non-increasing - $$3\not<3, 3\not<2, 2\not<1$$. The definition is similar for non-decreasing. (Note in particular that non-increasing does not imply decreasing, as the above example shows.)

The sequence $$4,1,2,3$$ has both increases ($$2$$ to $$3$$) and decreases ($$4$$ to $$1$$); so it is - awkwardly - not increasing, not decreasing, not non-increasing, and not non-decreasing.

Ugh!

• To add to the confusion in France (and possibly some other countries), we consider increasing as losely increasing while in U.S it means strictly increasing (the same way we consider positive as being $\ge 0$ and in U.S $>0$). Thus for me $3,3,2,1$ is decreasing.
– zwim
Commented Jul 22, 2020 at 20:13

My understanding is that a non-increasing sequence is different than a decreasing one in the sense that decreasing one "decreases every time" and non-increasing "doesn't increase at all" So the example you gave wouldn't be non increasing as it increases from 1 to 2 and 2 to 3.

There are three conditions of change to the next event. Increasing, decreasing and remaining constant. Negative to these three states are non-increasing, non-decreasing and not remaining constant, i.e, varying. So they are not mutually exclusive.