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How to Think About Analysis
Analysis (sometimes called Real Analysis or Advanced Calculus) is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared. It is not like other Analysis books. It is not a textbook containing standard content. Rather, it is designed to be read before arriving at university and/or before starting an Analysis course, or as a companion text once a course is begun. It provides a friendly and readable introduction to the subject by building on the students existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated formal versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs, pointing out typical areas of difficulty and confusion and explaining how to overcome these. The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses: it explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research-based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics.
274 pages, Kindle Edition
First published January 1, 2014
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Displaying 1 - 24 of 24 reviews
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December 14, 2023Why am I, an Astrophysics student, taking pure mathematics modules? Well dear reader there is only one answer, academic inflicted masochism.
July 10, 2019
Great book for students of mathematics who aren't accustomed to mathematics past the high school level and the rigour of analysis, this book is NOT an analysis book but it will provide you with the needed tools to be able to start reading more advanced mathematics. Alcock is a professor of mathematics whose research is geared towards teaching and uses that research to make pure math more accessible to everyone.
June 19, 2026
The only thing that could make this book better is if it covered more of analysis. I understand why it didn't, but I definitely found it harder to understand topics that weren't covered in-depth (or at all) in this book, because Lara's approach is so relatable and intuitive.
The whole (short) section on Taylor Series essentially boiled down to the idea that we use them to approximate functions, so remember that a function is equal to its Taylor series + some "leftover stuff" because we're approximating the function, so there's always a bit left over. Sounds super basic and superficial, and it is, but I truly took that idea to heart and had it in my head the whole time I was studying Taylor series and working through related problems in my exam.
I also really appreciated all the diagrams and intuitive descriptions of analysis terms and topics, because they helped me grasp the general idea of something before trying to make sense of the precise definitions, theorems, and formulas that were relevant for problem-solving. It's an approach that really worked for me, and I highly recommend this book to anyone who's preparing for (or already struggling with) studying real analysis.
The whole (short) section on Taylor Series essentially boiled down to the idea that we use them to approximate functions, so remember that a function is equal to its Taylor series + some "leftover stuff" because we're approximating the function, so there's always a bit left over. Sounds super basic and superficial, and it is, but I truly took that idea to heart and had it in my head the whole time I was studying Taylor series and working through related problems in my exam.
I also really appreciated all the diagrams and intuitive descriptions of analysis terms and topics, because they helped me grasp the general idea of something before trying to make sense of the precise definitions, theorems, and formulas that were relevant for problem-solving. It's an approach that really worked for me, and I highly recommend this book to anyone who's preparing for (or already struggling with) studying real analysis.
July 16, 2026
Enjoyability: ★★★★
The book is very readable and quite enjoyable modulo you being interested in real analysis. It can get a tad rough-going at times, but tough in the context of ‘a light read’, which is where I had pigeonholed it.
Enlightenment: ★★★
I am in the middle of studying analysis, so a lot of what’s in here is not new to me. The book suggests some strategies for facilitating learning of the subject, and I appreciate those.
Originality: ★★★★
The idea of making a very accessible introduction to the main ideas and strategies one needs for succeeding in an undergraduate course is pretty new: the traditional strategy in this regard is to give you a copy of the Little Blue Book (i.e., Principles of Mathematical Analysis) and throw you into the middle of the Mariana Trench. Luckily, there are alternatives to this (The Real Analysis Lifesaver, Understanding Analysis, and Real Analysis: A Long-Form Mathematics Textbook). What makes this volume distinctive is that it is not a textbook, rather, it plants some basic concepts and ideas as well as ways of handling them as part of a ‘tilling the soil’ approach for an analysis course.
Cultural weight: DA
This doesn’t really apply. Looking into the future, I can imagine it becoming very useful and a popular lifeboat for undergraduate students.
*Analysis is hard!*
Li Bai has a famous poem about how “the road to Shu is hard, harder than climbing the sky” (check my youtube channel for a recording of me reading and explaining it). You know what else is hard? Analysis. Not calculus! Calculus gets a bad rap, but when you take a look at it, it mostly boils down to calculating derivatives of simple functions with a simple set of rules you can memorize in an afternoon, and calculating integrals, which is messier and more of a bag of different tricks for different cases, but is ultimately quite doable. Now real analysis, that’s a completely different beast, and the first example you’ll find in your studies of what real mathematics looks like. Now you already know what is true, but you have to learn how to prove it rigorously, deriving results from axioms and theorems and checking examples of how it’s done. Also, real analysis is brutally, painfully counterintuitive: lots of things that seem natural and true do not apply to real numbers, and you’re going to learn this the hard way (that is, by spending more than an hour banging your head and attempting to show things that cannot be done). Even with proper assistance and support, this is going to be tough.
This is where this little book (222 pages of text, palm-sized) comes in handy. It will try to take you by the hand and guide you through a bird’s-eye view of the Land of Analysis. So buckle your seatbelt, Dorothy, ’cause Kansas is going bye-bye!.
*Contents*
The book is asymmetrically divided into two parts. The first one, “Studying Analysis”, which makes up about a third of the book and covers chapters one to four, gives you the most foundational baggage you need when you’re starting out in the topic: examples of notation and how to read it; what axioms, definitions, and theorems are; how to connect them to specific examples and/or graphs; how proofs are built; and a few tips on how to make the most of your analysis classes. It is mostly pretty basic stuff — perhaps too basic for the intended audience, who are likely, in my experience, to have had at least some “Introduction to Mathematical Logic and Proofs” course before landing in real analysis — but I never say no to a brief refresher.
The bulk of the book (two-thirds) consists of part two, “Concepts in Analysis” with six chapters followed by a very brief conclusion. This is the real bread and butter of the volume: in each chapter, you get an excellent, step-by-step, intuitive but also kind-of-rigorous-enough presentation of one big topic in real analysis. Chapter five is devoted to Sequences, chapter six to Series, chapter seven to Continuity, chapter eight to Differentiability, chapter nine to Integrability, and chapter ten to basic properties of the Real numbers. The blueprint for each of these chapters follows a similar pattern: we start with informal and intuitive approximations to the concept, with lots of graphs and examples and some common pitfalls, all building up to the precise definition of that concept. After that, we see some examples of where this takes us, with proofs or sketches of theorems that rely on the mathematical machinery that has just been explained, and that act as stepping stones towards the next concepts, steps, or machinery. The book doesn’t come with exercises, but it does rather frequently ask the reader to ponder and think about aspects of what has been explained, and to try to prove similar things to those in the volume. The book is also “modular”: the writer explains that you can read it in order, “as is,” or you can skip stuff you already know and/or jump to stuff that you are more interested in.
*Conclusion*
I was rather happy with this book and would wholeheartedly recommend it. It does precisely what it says on the package: it helps you prepare for the challenge of going through a real analysis course, gives practical advice, and tries to clarify, in a very intelligible way, the key concepts you need to understand to succeed. It is great as an introduction and also as a companion while you’re following a textbook.
I don’t have any serious issues to mention. As I mentioned before, some of the earliest material is pretty basic. I began reading it after having done some university-level math courses and while being not quite halfway (maybe a third of the way?) into Bartle and Sherbert’s Introduction to Real Analysis. My study of this textbook had already covered aspects like basic logic, sets and functions, induction, the properties of ℝ, sequences, and series, so the intuition and content about those topics were less helpful than those concerning the later topics, but that isn’t the book’s fault. I was somewhat surprised by Lara Alcock’s choice to place the properties of real numbers as the last chapter of the book. But this is very minor nitpicking. This is a great companion volume to real analysis, and you will probably profit a lot from reading it.
The book is very readable and quite enjoyable modulo you being interested in real analysis. It can get a tad rough-going at times, but tough in the context of ‘a light read’, which is where I had pigeonholed it.
Enlightenment: ★★★
I am in the middle of studying analysis, so a lot of what’s in here is not new to me. The book suggests some strategies for facilitating learning of the subject, and I appreciate those.
Originality: ★★★★
The idea of making a very accessible introduction to the main ideas and strategies one needs for succeeding in an undergraduate course is pretty new: the traditional strategy in this regard is to give you a copy of the Little Blue Book (i.e., Principles of Mathematical Analysis) and throw you into the middle of the Mariana Trench. Luckily, there are alternatives to this (The Real Analysis Lifesaver, Understanding Analysis, and Real Analysis: A Long-Form Mathematics Textbook). What makes this volume distinctive is that it is not a textbook, rather, it plants some basic concepts and ideas as well as ways of handling them as part of a ‘tilling the soil’ approach for an analysis course.
Cultural weight: DA
This doesn’t really apply. Looking into the future, I can imagine it becoming very useful and a popular lifeboat for undergraduate students.
*Analysis is hard!*
Li Bai has a famous poem about how “the road to Shu is hard, harder than climbing the sky” (check my youtube channel for a recording of me reading and explaining it). You know what else is hard? Analysis. Not calculus! Calculus gets a bad rap, but when you take a look at it, it mostly boils down to calculating derivatives of simple functions with a simple set of rules you can memorize in an afternoon, and calculating integrals, which is messier and more of a bag of different tricks for different cases, but is ultimately quite doable. Now real analysis, that’s a completely different beast, and the first example you’ll find in your studies of what real mathematics looks like. Now you already know what is true, but you have to learn how to prove it rigorously, deriving results from axioms and theorems and checking examples of how it’s done. Also, real analysis is brutally, painfully counterintuitive: lots of things that seem natural and true do not apply to real numbers, and you’re going to learn this the hard way (that is, by spending more than an hour banging your head and attempting to show things that cannot be done). Even with proper assistance and support, this is going to be tough.
This is where this little book (222 pages of text, palm-sized) comes in handy. It will try to take you by the hand and guide you through a bird’s-eye view of the Land of Analysis. So buckle your seatbelt, Dorothy, ’cause Kansas is going bye-bye!.
*Contents*
The book is asymmetrically divided into two parts. The first one, “Studying Analysis”, which makes up about a third of the book and covers chapters one to four, gives you the most foundational baggage you need when you’re starting out in the topic: examples of notation and how to read it; what axioms, definitions, and theorems are; how to connect them to specific examples and/or graphs; how proofs are built; and a few tips on how to make the most of your analysis classes. It is mostly pretty basic stuff — perhaps too basic for the intended audience, who are likely, in my experience, to have had at least some “Introduction to Mathematical Logic and Proofs” course before landing in real analysis — but I never say no to a brief refresher.
The bulk of the book (two-thirds) consists of part two, “Concepts in Analysis” with six chapters followed by a very brief conclusion. This is the real bread and butter of the volume: in each chapter, you get an excellent, step-by-step, intuitive but also kind-of-rigorous-enough presentation of one big topic in real analysis. Chapter five is devoted to Sequences, chapter six to Series, chapter seven to Continuity, chapter eight to Differentiability, chapter nine to Integrability, and chapter ten to basic properties of the Real numbers. The blueprint for each of these chapters follows a similar pattern: we start with informal and intuitive approximations to the concept, with lots of graphs and examples and some common pitfalls, all building up to the precise definition of that concept. After that, we see some examples of where this takes us, with proofs or sketches of theorems that rely on the mathematical machinery that has just been explained, and that act as stepping stones towards the next concepts, steps, or machinery. The book doesn’t come with exercises, but it does rather frequently ask the reader to ponder and think about aspects of what has been explained, and to try to prove similar things to those in the volume. The book is also “modular”: the writer explains that you can read it in order, “as is,” or you can skip stuff you already know and/or jump to stuff that you are more interested in.
*Conclusion*
I was rather happy with this book and would wholeheartedly recommend it. It does precisely what it says on the package: it helps you prepare for the challenge of going through a real analysis course, gives practical advice, and tries to clarify, in a very intelligible way, the key concepts you need to understand to succeed. It is great as an introduction and also as a companion while you’re following a textbook.
I don’t have any serious issues to mention. As I mentioned before, some of the earliest material is pretty basic. I began reading it after having done some university-level math courses and while being not quite halfway (maybe a third of the way?) into Bartle and Sherbert’s Introduction to Real Analysis. My study of this textbook had already covered aspects like basic logic, sets and functions, induction, the properties of ℝ, sequences, and series, so the intuition and content about those topics were less helpful than those concerning the later topics, but that isn’t the book’s fault. I was somewhat surprised by Lara Alcock’s choice to place the properties of real numbers as the last chapter of the book. But this is very minor nitpicking. This is a great companion volume to real analysis, and you will probably profit a lot from reading it.
December 13, 2021
I have read her two main books, including this one. Her math and writing style are perfect. This book provides some practical insights on how to study advanced math, in particular the last few pages would be very helpful:). For me, even though it is quite banal, her advice on always writing my own denifition list in any math course was a turning point in substantially improving a material understanding. Apart from this, this book is a math book with many proofs quite deeply explained. Topics of sequences, series, continuity, and reals are covered well. In sum, there are very few such math books out there that, on the one hand, are a real math book demanding a good math level, but at the same time, are written by an author that wants to make every theorem or math fact in a book intuitively understandable as well. Sometimes she succeedes
September 4, 2015
This is fantastic little book that will help you get rid of that instinctive fear of Real Analysis (those epsilons, deltas and countless mathematical symbols). The book is short but very engaging with very good diagrams and explanations. Analysis is hard but it is also elegant, clever and very rewarding. This book goes a long way to easy your path in its study ("pointing out typical areas of difficulty and confusion and explaining how to overcome these").
Topics covered: sequences, series, continuity, differentiability, integrability and the real numbers.
Topics covered: sequences, series, continuity, differentiability, integrability and the real numbers.
October 5, 2017
I have plenty of time on the train each morn and aft and I have been using this time to revisit analysis. It doesn't elicit the terror it did as an undergrad in fact I really enjoyed reading this and going through the proofs she cites. Fun Stuff... And before you know it. You're at work with your brain humming.
August 28, 2021
Lovely read. Motivated me enough that I might, someday, go down the path of formally pursuing Analysis. In general also it's a great book to improve mathematical literacy. Gives a neat birds eye view of Analysis, makes you comfortable into reading mathematical statements and proofs, and also presents excellent tips on how to structure your study for this as well as other mathematical domains.
June 20, 2022
I never felt easy with the limiting process involved in Calculus. To remedy that I wanted to teach myself Analysis. However I got intimidated by the sheer amount of logical symbols in a typical passage in a typical Analysis book.
I found this book to be a good friend. I could follow it well. It gave me ammunition and a mindset to tackle further Analysis.
I found this book to be a good friend. I could follow it well. It gave me ammunition and a mindset to tackle further Analysis.
August 20, 2025
8.2025 - I have to reread this masterpiece
Best book on Real Analysis. Seriously!!! Universities need to adopt this book as their textbook, please. At least replace recommending Introduction to Real Analysis by Bartle, Robert G. to this
Best book on Real Analysis. Seriously!!! Universities need to adopt this book as their textbook, please. At least replace recommending Introduction to Real Analysis by Bartle, Robert G. to this
October 23, 2021
I really get it (and i'm not math student at all). This book can be example for all other math study books. Easy examples, easy explanations and best introduction into theme (even some humor), this things it's a pillars of good book for me
If you think it's hard:
-
∀a, b ∈ R, a + b = b + a;
∃ 0 ∈ R s.t. ∀a ∈ R, a + 0 = a = 0 + a.
-
or
-
f : X → R is bounded above on X if and only if
∃ M ∈ R s.t. ∀x ∈ X, f(x) ≤ M.
-
You just don't readed second chapter of this book, where you can have all definitions of RA things
If you think it's hard:
-
∀a, b ∈ R, a + b = b + a;
∃ 0 ∈ R s.t. ∀a ∈ R, a + 0 = a = 0 + a.
-
or
-
f : X → R is bounded above on X if and only if
∃ M ∈ R s.t. ∀x ∈ X, f(x) ≤ M.
-
You just don't readed second chapter of this book, where you can have all definitions of RA things
August 19, 2024
I read this to prepare for grad level analysis and refresh my knowledge but I am HEAVILY wishing I had read this last year to prepare for my undergraduate course! The author is super knowledgeable and gives tips on how to transition into the class. I really like all of the diagrams and the conversational style ( something that's often missing from Analysis materials ). She also talks about why students might misunderstand or misinterpret concepts and why! I will definitely recommend this book to any undergrads anxious about taking analysis.
February 28, 2026
Yazarın diğer kitapları gibi bu kitabını da akademik seviyede matematik öğrenmeye yeni başlayan birisi için yazılmış harika bir kaynak ve yol gösterici, düşünce dünyasını şekillendirici bir kaynak olarak görüyorum. Başka hiç kimsenin değinmediği şekilde sohbet havasında, yapılan genel hataları açıklayıp ispat mekanizmasının nasıl çalıştığına dair örnekler veriyor. Sadece şu an yoğun matematik çalışma dönemimde olmadığım için ara veriyorum. Elime kâğıt kalem alarak kitabın başına tekrar oturacağım.
October 4, 2020
I only read the first part for practical reasons (for now), and I'm blown away. This book is a must-read for anyone embarking upon a challenging study. Even beyond mathematics this book is relevant because it teaches you how to tackle the course material and inoculates you from your brain's self-sabotaging tendencies. Looking forward to part two.
April 29, 2019
I have feared analysis since studying it at first-year undergraduate level when I didn't really get it at all. This book makes it seem easy, and even enjoyable. A great way to prepare to study this topic.
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August 27, 2020Gentle introduction to analysis with valuable advice for almost any reader.
September 11, 2021
Good intro but needs more depth or examples of how to write proofs.
March 30, 2023
Very readable and motivating
Read
August 20, 2023A useful resource
June 12, 2024
As with all nightmares this one had an ending.
November 11, 2024
I wish I had this book when I studied Analysis in university!
Lara explains concepts and ideas so well, it's a pleasure read and follow along.
Lara explains concepts and ideas so well, it's a pleasure read and follow along.
January 13, 2018
Maybe I got too excited after my Analysis 3 course, I overestimated the level of this book it is actually basic first year university material. I skimmed it and I think it written was very clearly and nicely.
January 20, 2017
This book was extremely helpful to me while I was taking a Graduate level Analysis. It served as an extra resource I could go to in order to better understand the material. I only gave four stars because I found the book lacking in some of the concepts I struggled with like the theorems involving open, closed, compact, etc. It is possible I struggle with them because they were thrown in before the midterm, but it still would have been helpful to have information in the book about them. I do understand that not everything can be covered in books. At the same time, my course did not cover sequences and this book had an entire chapter on them. I was most intrigued about the sequences chapter and tried to read it anyway, but I ended up just skimming it because I was already finished with my class and wanted to move on to other reading. I hope to delve into the sequences chapter further at some points. When I do, I will update this review to reflect that. An overall very good book to give some good foundation to my Analysis grad class.
August 1, 2015
Quite superb. Compulsory reading for any nascent maths undergraduates.
Displaying 1 - 24 of 24 reviews