Read-along: Rudin's PoMA: Week 1 (1st April to 7th April)

Goal: The goal for this week is to define $ \mathbb{R} $ axiomatically, as an ordered field satisfying the Archimedean property.

Reading: Chapter 1 of Rudin, except the appendix. (So pages 1 to 17, up to and including paragraph 1.38.) The chapter is scanned here. Next week we will read the appendix in detail, but there is no point bothering to construct the real numbers from scratch before we understand what we want!

The important theorems are:

The existence of $ \mathbb{R} $ (!), theorem 1.19 (proof postponed to next week)
The Archimedian property, theorem 1.20

Suggested problems: Exercises for Chapter 1, from page 21 onwards: problems 1-7 this week, as well as a nice variety of the supplementary problems below that. There is no rush, we will spend at least next week on this chapter as well.

It would be nice if everyone wrote up their solution to one or two of these problems so that we can collect them all online here and learn from each other ($ \LaTeX $ not required, just scan it if you want to as long as it’s readable!). If we end up having a seminar meeting this week then these solutions are something we can discuss there.

Additional exercises, easier than and supplementary to Rudin: Here are a few easier exercises with one or two line solutions, as well as some examples (as Rudin does not include too many). solutions

There are a couple of facts about (ordered) fields which we want to use but Rudin does not prove. Using only the material developed up to (and including) proposition 1.14, prove that if $ x $ and $ y $ are field elements then $ -(x + y) = (-x) + (-y) $.
If $ S $ is a set, $ x \in S $ is called the maximum of $ S $ (and we write $ x = \max S $) if for every $ y \in S $, $ y \leq x $.
- Write a reasonable definition for the minimum value of a set $ S $.
- If $ S \subseteq \mathbb{R} $, define $ -S $ by $-S = \{-s : s \in S\}.$ Show that $ -S $ has a minimum iff $ S $ has a maximum.
- How are maxima, minima, suprema, and infima related?
Find the supremum and infimum of the following sets (you may use all the standard results you know of, including the properties of $ e^x $, without proof). Which have a maximum or a minimum?
- $ A = \{a + a^{-1} : a \in \mathbb{Q}, a > 0\} $
- $ B = \{a + (2a)^{-1} : a \in \mathbb{Q}, 1/10 \leq a \leq 5 \} $
- $ C = \{ xe^x : x \in \mathbb{R} \} $
Let $ A $ and $ B $ be bounded above subsets of $ \mathbb{R} $. Define $ (A + B) := \{ a + b : a \in A, b \in B \} $. What can you say about $ \sup (A + B) $?
Prove that, if $ x $ and $ y $ are real, then there exists some $ \xi \in (x,y) \setminus \mathbb{Q} $. (Hint: there is a very elegant proof of this; one ingredient is the Archimedian property, theorem 1.20; the other is the fact that $ p\sqrt{2} $ is irrational for all rational $ p $.)
Do there exist $ x $ and $ y $ real such that:
- $ x $ and $ y $ are rational but $ x^y $ is irrational?
- $ x $ and $ y $ are irrational but $ x^y $ is rational?
Which subsets of $ \mathbb{Q} $ have an inf or a sup?
Define . Show that $ \mathbb{Q}[\sqrt{2}] $ is:
- a field;
- which contains $ \mathbb{Q} $; and
- such that there exists $ \xi \in \mathbb{Q}[\sqrt{2}] $ satisfying $ \xi^2 = 2 $.
- Does $ \mathbb{Q}[\sqrt{2}] $ satisfy the least-upper-bound property?
The previous exercise gives us a field $ F $ satisfying $ \mathbb{Q} \subset F \subset \mathbb{R} $ (where the set inclusions are strict, i.e. they are not equalities). Does there exist a field $ G $ such that $ G \subset \mathbb{Q} $ (where the inclusion is again strict)?
Prove (without using theorem 1.20) that, for every $ r, s \in \mathbb{Q} $, then there exists $ n \in \mathbb{N} $ such that $ n r > s $. (This is the Archimedian property for $ \mathbb{Q} $.)
Using the material developed up to (and including) theorem 1.37, prove that if $ k > 0 $ and if $ \mathbf{x} \in \mathbb{R}^k $ then there exists $ \mathbf{u} \in \mathbb{R}^k $ such that $ | \mathbf{u} | = 1 $ and $ \mathbf{u} \boldsymbol{\cdot} \mathbf{x} = | \mathbf{x} | $.

Some advice: A mostly true, if impractical, piece of advice:-

So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting yourself to the reading, and solving the exercises till you know it by heart. Unfortunately, I suspect the reader is looking for advice on how not to read, i.e. what to skip, and even better, how to read only some isolated highlights. (From Saharon Shelah, “Classification Theory and the Number of Non-Isomorphic Models”; quoted in Just and Weese, “Discovering Modern Set Theory I”; quoted on MSE.)

A more practical piece of advice:

Don’t be upset if Rudin gives an argument that’s really really clever and it seems like you can’t see where he got it from. Sometimes the argument isn’t important. See for example how he shows that the set of rationals whose square is less than 2 does not have a least upper bound in $ \mathbb{Q} $. (Reddit)

Next: Next week the plan is to read through the construction of $ \mathbb{R} $. Then we will move into chapter 2, which is the longest slog in the book; so despite that chapter being only 25 pages, I expect we will spend four or so weeks on it.