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.
 A: 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!
A: 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. 
A: 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.
