Monotonic
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I made this just to make it blue
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This page requires references, it is on a todo list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
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Find an order theory book, also I think that huge category theory PDF (Harold Simmons) has it
Definition
A map, [ilmath]f:X\rightarrow Y[/ilmath] between two posets, [ilmath](X,\sqsubseteq)[/ilmath] and [ilmath](Y,\preceq)[/ilmath] is monotonic or monotone if:
 [ilmath]\forall a,b\in X[a\sqsubseteq b\implies f(a)\preceq f(b)][/ilmath], or in words:
 It preserves the ordering.
For a sequence
Recall that a sequence, [ilmath] ({ A_n })_{ n = 1 }^{ \infty }\subseteq X [/ilmath] (for some poset, [ilmath](X,\sqsubseteq)[/ilmath]) can be considered as a mapping:
 [ilmath]A:\mathbb{N}\rightarrow X[/ilmath] given by [ilmath]A:n\mapsto A_n[/ilmath]
We can now apply the above definition directly.
Work needed
TODO: These
 How can we have monotonically decreasing things? Via the dual partial ordering of course! To have [ilmath]\le[/ilmath] is to induce a unique [ilmath]\ge[/ilmath]  these are distinct orderings.
 Not sure, but probably some call this isotonic, while monotonic is either increasing or decreasing.
 Unite with monotonic set function
References
