Name: Real Analysis
ISBN: 9781724510129

Real Analysis

This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 100 illustrations. The writing is relaxed and includes periodic historical notes, poor attempts at humor, and occasional diversions into other interesting areas of mathematics.The text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis.The author believes most textbooks are extremely overpriced and endeavors to help change this.

What Hacker News says

seanhunterJun 2026

One of my favourite dedications is in the preface to “Real Analysis: A long-form mathematics textbook” by Jay Cummings (which is as it sounds a maths textbook). The dedication reads

“To my loving wife, who read this entire book, apart from the maths parts.”

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beryilmaJul 2024

As a PSA for anybody who wants a very readable introduction to "real" math, check out Jay Cummings' two books: Proofs and Real Analysis at https://longformmath.com/

Each paperback book costs less than $20 on Amazon.

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gsinclairJul 2024

Something more talky is Real Analysis by Jay Cummings. Great book!

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PatentlyDC123Apr 2024

Real Analysis. I know Baby Rudin is probably the gold standard, but Jay Cummings' Proofs and Real Analysis have been great for people like me delving deeper into mathematics.

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beryilmaApr 2024

One the same subject and as accessible, I love the two books from Jay Cummings: Proofs and Real Analysis. Each just $16 on Amazon. It is a joy to read these books and try some of the exercises. I wish PDF versions were also available...

For those interested: https://webpages.csus.edu/jay.cummings/Books.html

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whatever1078Sep 2021

You need to transition to learning math.

One possible starting point is Book of Proof by Richard Hammach[0]. It's free. It will show you the rudiments of how math is actually done and thought of. Another good choice is Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang. These books have no prerequisites(good to go if you know how to do arithmetic with fractions/decimals; not that these are strictly necessary, rather that's about the level of comfort with math one should posses). Search on Amazon and you'll see there are a ridiculous number of books on this subject of all varieties of ways of presentation. Almost every week a new book pops up, it seems. Note, intro-to-proofs books are not only for novices, though. There are kinds written for advanced undergrads and beginning grad students as well. For example: Mathematical Concepts by Jugen Jost. This speaks to the wealth of the kinds of math books available.

Concurrently and/or after that you can study introduction to any of linear algebra, discrete math, group theory, analysis of reals, combinatorics, number theory, graph theory, algorithms, probability, category theory...whatever. Each of these are very deep subjects and there exist hundreds(if not 1000s of books from the past, current and future) of every imaginable level, depth and presentational quirk. Again check Amazon for titles. Check libge*n for actual files.

The following intro books are so easy anyone and they momma can read these with utmost ease:

Discrete Math by Susanna Epp

How to Think about Analysis by Lara Alcock

Linear Algebra: Step by Step by Kuldeep Singh

Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings

A Tour Through Graph Theory by Karin Saoub

The above mentioned books are a few examples of elementary books that you can get started with right this minute.

Note and remember the wealth of possible books, though. For example, consider the subject of math analysis.

We can break it into three general parts: real analysis, complex analysis, functional analysis. Each can be further taken apart into measure theory, vector spaces, topology etc. Further, topology alone branches out into point-set topology, algebraic topology, differential topology etc. Each of these fly under many different names. Say, vector spaces is a huge part of linear algebra. For the beginners, it's usually the finite dimensional vector spaces (aka the title of a famous linear algebra book) version whereas the infinite version will take you into func analysis. Vector spaces are also usually included as a chapter in books on abstract algebra as they are an algebraic structure just like groups, rings, modules. But if there is linear algebra, there must be non-linear algebra too. Well, yes. Roughly, it corresponds to what's called algebraic geometry. It never ends.

The point is every topic in math has loads of books dedicated to it. In turn every topic has a (sub)^{n}topic and there are tons of books on those. Each of the books come in elementary, quirky, armchair enthusiast, average undegrad student, middling grad student, kickazz graduate math aficionado, researcher, undergrad researcher wanna-be etc levels. There are also olden books, classics, those just published (say, at most 5 years ago) and up-and-coming ones. Also, each term in math has bagillion synonyms. So multiply the number of books above by n. I am not even gonna mention books written in different languages and lecture notes that are available for free online. A lot of published book often spend years as a free set of lecture notes before ending up on Amazon as a newly published book. There are also Olympiad books, proof compilations, capstone compilations, intro-to-research books, those that prove a single theorem and in the process go through tons of disparate math etc Take advantage of all of this.

[0] https://www.people.vcu.edu/~rhammack/BookOfProof/

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