Correct usage of the phrase "In the sequel"? History? Alternatives? While I feel quite confident that I've inferred the correct meaning of "In the sequel" from context, I've never heard anyone explicitly tell me, so first off, to remove my niggling doubts: What does this phrase mean?
(Someone recently argued to me that "sequel" was actually supposed to refer to a forthcoming second part of a paper, which I found highly unlikely, but I'd just like to make sure. )
My main questions: 
At what points in the text, and for what kinds of X, is it appropriate to use the phrase "In the sequel, X" in a paper? In a book? Is it ever acceptable to introduce definitions via 
"In the sequel, we introduce the concept of a "blah", which is a thing satisfying ..." 
at the start of a paper or book without a formal
"Definition. A "blah" is a thing, satsifying ..."
in the main text of the paper or book? Finally, out of curiosity, I'm wondering how long this phrase has been around, if it's considered out of date or if it's still a popular phrase, and what some good alternatives are.
 A: Your interpretation of how "in the sequel" is used in mathematical literature is correct: it means "in what follows", "in the remainder of the present text", "from now on"...As you can see, there are many other such expressions, and I don't think that "in the sequel" has any particular nuance of meaning that these other phrasings lack.  
On to your question of whether this is acceptable usage in a math paper: acceptable, yes, but not completely recommended.  In more detail:
Ths phrase is, as you point out, rather common, so any experienced reader of math will have seen it before.  Also its meaning should be relatively transparent to a literate native speaker of English: this is after all what the dictionary says that "sequel" means.
However, there are two drawbacks.  First, as Mariano says, nowadays we hear "sequel" used most commonly for the next movie in a franchise, so its use in a math paper will inevitably make some readers think that you are referring to [TITLE OF YOUR PAPER] Part II: This Time, It's Personal, or whatever.  Second, in my opinion it is somewhat hackneyed language and an idiomatic usage that doesn't add any meaning.  If you read enough math papers you'll find that there are certain linguistic ticks that people pick up from each other (and also sometimes, from writing in other languages), e.g. beginning a sentence with "Remark that..." sounds a little stilted and is probably a semiconscious translation of the french phrase "Remarquer que" (which I would translate as "Notice that").  Most people agree that good mathematical writing is as stylistically unobstrusive as possible, so using phrases that make people ponder and ask questions about them on websites is, arguably, slightly too distracting.  
Anyway, it's  no big deal.  If I were a referee or even a copyeditor, I would almost certainly let the author use "in the sequel" if she wants.
A: I would never write that (mostly because it really sounds like the definition will be in another paper...). I'm pretty sure I'd write «In what follows, ...».
A: Mathematics is generally seen as a precise and specific language.  Surely if there is ambiguity in the phrase "in the sequel" it should be avoided.  There are far more accurate ways of expressing both the meanings that phrase could carry so why create needless confusion?
A: I've taught math for more than a decade at the college level, and I have been a professional copy editor for almost a decade. I find this phrase distracting and potentially confusing or misleading, so my recommendation is to avoid it. I found it in a paper about "communication for power distribution grid," and I have substituted it with a standard, non-ambiguous phrase. Finally, none of Merriam-Webster's definitions of the word "sequel" justify this phrase with the intended meaning.
A: I've seen this phrase for years and am just now learning that it does not refer to a subsequent paper or book. This has to be the dumbest phrase in the history of mathematical writing. "In the sequel," considered harmful, indeed. Unless you are authoring a paper titled, "A model-theoretic approach to NP-hard approximation algorithms Part Deux, Electrical Boogaloo," please do not use this phrase.
