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I am a statistician who wishes to learn real analysis in order to better understand the foundations of statistics. With that aim in mind I plan to go through Rudin's classic on "Principles of Mathematical Analysis".

Given the above context can I skip chapter 1? It seems to me that the material in chapter 1 is not as important to someone with my goals. I understand that it may help in establishing the need for rigor in mathematical arguments but that is something I already appreciate. In particular, I wonder will skipping chapter 1 impede my understanding of the material in subsequent chapters?

Any advice will be appreciated.

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You should at least read it, if not do the exercises. –  Qiaochu Yuan Aug 30 '10 at 0:15
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It probably won't, but it really is nice from a mathematical perspective to see how the reals are actually constructed (even though there is much to be said for the Cauchy sequence choice). –  Akhil Mathew Aug 30 '10 at 0:44
    
I know I've made this plug before, but a nice alternative to Rudin's chapter 1 is Terrence Tao's Honors Analysis notes week 1 and 2 (roughly). His development is more detailed and avoids Dedekind cuts. –  WWright Aug 30 '10 at 2:24
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@WWring, why is "avoiding Dedekind cuts" a plus? –  Mariano Suárez-Alvarez Aug 30 '10 at 3:25
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It's really nice to see how real numbers are actually constructed from a mathematical point of view. But, for anyone outsice the mathematical kingdom, this can be skiped with no harm. In fact, I have an anecdote from my youth: we were exposed to that construction the first day we begun maths. No one understood anything and rigth now real numbers are no more explained as the limit of Cauchy sequences of rational numbers in my University. –  a.r. Aug 30 '10 at 12:03

4 Answers 4

up vote 16 down vote accepted

I'll go through the paragraphs of the third edition, motivating why you should/shouldn't consider them

INTRODUCTION

You should carefully read this. This paragraph is not so important for the subsequent development, but it's fundamental for your understanding of the utility of $\mathbb{R}$, it contains an enlightening example wich shows you (with some weird algebraic trick) that $\mathbb{Q}$ contains gaps and so we really need to "patch" it in order to do interesting things.

ORDERED SETS

You should read only the definitions of bound and least upper bound (or supremum). They really matter and are used, explicitly or not, in many theorems. In particular, the latter is subtle and you should practice with it, for example with exercises 4 and 5 at the end of the chapter. The rest of the paragraphs regards ordered fields, you can skip it if you are used to work with $\mathbb{R}$ and its ordering, and this practical experience should suffice to you. Perhaps, you could find strange and interesting that $\mathbb{C}$ cannot be ordered without destroying its algebraic properties.

FIELDS

You can skip this. If your interest is real analysis, your only field will be $\mathbb{R}$ (perhaps $\mathbb{C}$ sometimes) and as said above, the practical properties of field and ordering should be enough for you (e.g. if $a,b,c \in \mathbb{R}$ and $a < b$ then $a + c < b + c$, you cannot divide by zero, etc.)

THE REAL FIELD

There are two ways to build $\mathbb{R}$. The first is axiomatic: you say "how I'd like to work in place that has such property" and magic! You have it by axiom. The second way is contructive: you take $\mathbb{Q}$, do something on it and come up with a mathematical structure that act as $\mathbb{R}$, has the properties of $\mathbb{R}$ and you eventually call it $\mathbb{R}$. It's very subtle and not practically useful, you should skip the latter method, reported in the appendix of the chapter, and know that when you follow the axiomatic method your are speaking of something that exists, in some mathematical sense. You should also consider theorem 1.20 (archimedean property and density of $\mathbb{Q}$ in $\mathbb{R}$) if you don't read the proof, at least carefully read the statement, it's very used and justify some mysterious things as: if $a \in \mathbb{R}$ and $0 \leq a < \epsilon$ for all $\epsilon > 0$ then $a = 0$. Jump over the existence of the n-th root of a positive real, it's intuitive and you can prove it later in different (and simpler) ways.

THE EXTENDED REAL NUMBER SYSTEM

Not only it's not very useful, I think it's dangerous to introduce symbols for infinity when someone still isn't completely conscious of what infinity is and how it acts in many theorems of analysis. Skip.

THE COMPLEX FIELD

As in the case of the real field, if you know what $\mathbb{C}$ is and how to work with it, you can safely skip this, or read it later if you need. The only thing that you probably need are the trianglular inequality in theorem 1.33 and theorem 1.35, known as Cauchy–Schwarz inequality.

EUCLIDEAN SPACES

Read it, it's used in the following chapters.

APPENDIX

As said above, skip it.

EXERCISES

As said above, 4 and 5 are very useful. I also suggest you to work on 6 and 7, they teach you what we mean when we say things like $3^{\pi}$ or when we talk about logarithms.

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This is very helpful. Thanks. –  sxv Aug 30 '10 at 13:01
    
+1: Great answer! –  Aryabhata Aug 30 '10 at 14:22
    
What a fantastic answer. –  alexx Aug 30 '10 at 19:24

It is probably less work to read through the first chapter than to ask if reading through it is necessary! Specially in the context of you planning to read through the whole book.

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From a strategic point of view: given that mathematical terminology and formalism are not totally standardised, it's worth at least reading through the chapter quickly, at least to stop yourself later having to refer back to what the various terms and symbols mean.

This advice goes for any book, I guess.

From the point of view of content: no one but you is in a position to really decide whether you understand the material enough to skip the chapter.

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This is a particularly relevant comment with respect to Rudin, since some of his terminology and notation are at variance with modern conventions. For instance, what he calls a "neighborhood" is what everyone else calls a "ball". –  Nate Eldredge Aug 30 '10 at 17:36

I think you answered your own question. If you are willing to accept that the reals are the unique totally ordered archimedean field with the least upper bound property, then OK skip it. But you may wish to refer back to it to know what all of these adjectives mean.

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