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I know we define the reals as the equivalence classes of Cauchy sequences, but here the author is trying to motivate this and I'm not totally following how. Here is my summary of what he's saying. For context, right before this section, we proved that a Cauchy sequence is a convergent sequence.

Summary: First, he says that we should add to the rationals the limits of a Cauchy sequence of rationals which we know to be a real number. But, we don't know if this is enough yet for $\mathbb R$ since he didn't define $\mathbb R$ yet. So, he raises the possibility of Cauchy sequences of irrational numbers and ponders whether or not they converge to a "new variety of real numbers." It seems not because the Cauchy sequence of reals has the same limit as Cauchy sequence of rationals, and he gives a proof. So basically what we can conclude up to this point is that we should add to the rationals the limits of a Cauchy sequence of rationals (this accounts for the limits of a Cauchy sequence of irrational numbers, hence the reals in general, which is confusing bc we're trying to define the reals and how do we know the reals consist of the irrationals and rationals before defining it??)

Anyway, he says "Another informal demonstration that we should get every real number as a limit of Cauchy sequences of rational numbers is the infinite decimal expansion." Hold up, this is where I'm confused because "another" suggests that we already showed EVERY real number is a limit of Cauchy sequences of rational numbers, but instead we showed that every Cauchy sequence converges to a real number. I don't think these are the same. Moreover, the author didn't yet define at this point in the book that the reals = equivalence classes of Cauchy sequences. So I'm quite confused.

He then goes on to explain an actual explanation for "every real number is a limit of Cauchy sequences of rational numbers": if we assume a real number $x$ has an infinite decimal representation, then we can use that representation as a Cauchy sequence.

Can someone paraphrase what the author is saying as well as explain where the first explanation of "every real number is a limit of Cauchy sequences of rational numbers" is? I'm confused because this "intuitive"/"informal" discussion is just making me more confused.

Here's a link to the textbook: https://vdocuments.mx/strichartzthe-way-of-analysis-2000.html

It's page 33-34. I also attached photos below.

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enter image description here

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  • $\begingroup$ To be honest I find the exposition confusing instead of motivating. If a textbook is serious about explaining construction of reals they better explicitly tell readers to forget any ideas of reals they may have had and start from scratch with rationals. $\endgroup$
    – Paramanand Singh
    Sep 9, 2021 at 5:56
  • $\begingroup$ Also the construction via Cauchy sequences is really technical and involves ideas which require familiarity with notions like limit and it is suited for people who have already taken some calculus or analysis course. A much simpler approach is via Dedekind cuts which does not involve any machinery and instead uses the order relations of rationals. $\endgroup$
    – Paramanand Singh
    Sep 9, 2021 at 5:59
  • $\begingroup$ On a side note, unless the book is out of copyright it is illegal to share it online unless authorized by the publisher. You have already given the relevant excerpt from book and it is better to remove the link for book. $\endgroup$
    – Paramanand Singh
    Sep 9, 2021 at 6:06

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Let me ignore for a moment the idea of defining the real numbers.

In the theory of analysis (and of metric spaces), when you are talking about sequences, Cauchy sequences, and limits, it is extremely important that you talk about what space the sequence is in, and what space the limit is in. Now if the real numbers $\mathbb R$ are already regarded as known, and if the rational numbers $\mathbb Q$ have been embedded in $\mathbb R$ in the proper fashion, then one can ponder that situation and think about Cauchy sequences like this. It is not true that every Cauchy sequence in $\mathbb Q$ is convergent in $\mathbb Q$. But because every Cauchy sequence in $\mathbb Q$ is also a Cauchy sequence in $\mathbb R$, it follows that every Cauchy sequence in $\mathbb Q$ is convergent in $\mathbb R$. These are all theorems of analysis, true once you already know what the real numbers are.

Once you understand those thoughts, you can then back up to the problem of defining the real numbers, assuming that the rational numbers $\mathbb Q$ have already been defined. Consider the Cauchy sequences in $\mathbb Q$. Define the equivalence relation amongst Cauchy sequences in $\mathbb Q$. Consider the set of equivalence classes of Cauchy sequences in $\mathbb Q$. Define the sum and product operations on the set of equivalence classes of Cauchy sequences in $\mathbb Q$, and define the inequality relations on the set of equivalence classes of Cauchy sequences in $\mathbb Q$. Using the sum and product operations and the inequality relations, prove that the set of equivalence classes of Cauchy sequences in $\mathbb Q$ satisfies the basic properties of the real numbers. You're done:

The set of equivalence classes of Cauchy sequences in $\mathbb Q$ is the real numbers.

Finally, let me point out that your opening assertion is incomplete, and therefore nonsensical, as it stated: "we define the reals as the equivalence classes of Cauchy sequence". That's not correct. What's correct is that we define the reals as equivalence classes of Cauchy sequences in the rational numbers $\mathbb Q$.

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