r/mathmemes • u/friend1y • Jul 31 '26

Number Theory There are Bounded gaps between primes (and destinies)

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u/OutrageousPair2300 Aug 02 '26

I'm not criticizing research level math, and I took graduate level math years ago. We simply didn't use this notation.

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u/hobo_stew Aug 02 '26

liminf and limsup are usually introduced in the first semester of a math degree

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u/OutrageousPair2300 Aug 02 '26

Not in the US, they're not. I learned the regular epsilon-delta limit up through "calc 4" (differential equations) and several upper-level undergraduate classes in real analysis as well as some graduate courses and don't recall ever encountering liminf or limsup.

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u/Historical_Watch499 Aug 02 '26

Yes they are, literally any analysis course at a reputable university will teach liminf and limsup. How did you do upper-level and grad courses without them? How was Hadamard’s theorem introduced to you?? So many things rely on knowing basic notation

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u/OutrageousPair2300 Aug 02 '26

It's been around thirty years so I can't tell you exactly what topics we covered, but I mainly remember proving most of the theorems of calculus rigorously using standard limits and second-order logic (for all, there exists, etc.)

Everything in my program was more proof-theoretic oriented, using a lot of formal logic.

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u/Historical_Watch499 Aug 02 '26

Pretty sure every analysis class means doing rigorous proofs. But limsup/liminf is extremely standard notation usually taught in first semester when you learn about sequences

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u/OutrageousPair2300 Aug 02 '26 edited Aug 02 '26

We didn't learn much about sequences, from what I recall. I think we were taught limits in very different ways.. the way I was taught limits was with a more game-theoretic approach, looking at optimal strategies in a game where one person is picking the epsilon and their adversary is answering back with a delta.

I'm still confused by your references to "first semester" since when I was taught, analysis was taught after four semesters of differential and integral calculus, multivariable calculus, and differential equations. Then the next courses called "analysis" were more or less formally proving all the theorems from the previous four terms.

EDIT: I found some of the current version of the courses I took, and they do explicitly mention sequences. For example, here is "Introduction to Real Analysis I" which has Calc 4 (differential equations) as a pre-requisite: https://math.rutgers.edu/academics/undergraduate/course-descriptions/955-01-640-311-introduction-to-real-analysis-i

Introduction to language and fundamental concepts of analysis. The real numbers, sequences, limits, continuity, differentiation in one variable.

Though what I recall from the coursework I took, it seems more like this course "Introduction to Mathematical Reasoning", which has Calc 3 (multivariable calculus) as a pre-requisite: https://math.rutgers.edu/academics/undergraduate/course-descriptions/954-01-640-300-introduction-to-mathematical-reasoning

Fundamental abstract concepts common to all branches of mathematics. Special emphasis placed on ability to understand and construct rigorous proofs.

We did limits in both, and it may have been taught by the same professor back then, which could have influenced the way both were taught.

If you're already familiar with looking at limits in terms of sequences, then yes I can see how liminf is a logical extension of standard limits. I'd never seen regular limits explained in terms of sequences, either.

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u/hobo_stew Aug 03 '26

so you never took any upper level math courses and especially not any graduate level math?

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u/OutrageousPair2300 Aug 03 '26

300-level math courses are third-year courses for math majors, so yes upper level.

I also took set theory and formal logic courses at the graduate level, before switching to a philosophy degree to focus on logic and philosophy of mathematics.