Is a real number the limit of a Cauchy sequence, the sequence itself, a shrinking closed interval of rational numbers, or what?

I've been studying a collection of analysis books (one of them Bishop's Constructive version) and contemplating the reals. Correct me if I'm wrong, but I feel that I have seen the Cauchy sequence itself in some places and its limit in other places described as the real number. I do understand that nested closed intervals have a point as the limit of their countably infinite intersection. I can visualize an arbitrarily small interval of rationals, any of which has an equal claim (its seem to me, at least until other considerations are brought in) to being an "approximation" of this point. Unless we know the limit point (via the geometric series, for instance), what are we approximating if not a yet better approximation of a yet better approximation? (Maybe the "approximation" terminology works better for cuts.) I suppose I'm asking not only for the clarification of the dominant convention but also for "real talk" about the mathematical imagining of real numbers.

Solution 1:

Real numbers are the "equivalence class" of Cauchy sequences of rational numbers. You take the set of all Cauchy sequences of rational numbers and put an equivalence relation on it. When do you say two Cauchy sequences of rationals say $x_{n}$ and $y_{n}$ are related? They are related if for every given $\epsilon > 0$ ($\epsilon$ is rational here), there exists a $n_{0}$ such that $n > n_{0}$ implies

$$ |x_{n} - y_{n}| < \epsilon $$.

Now you say that each equivalence class is a real number. How do you see the reational numbers inside real numbers then? So, suppose $r$ is a rational number, then the constant sequence $x_{n} = r$ is a Cauchy sequence and the equivalence class it belongs to is the rational number $r$ in the set of real numbers. This is the identification of rationals inside the reals.

This is one way of thinking about the real numbers. The other way is through Dedekind Cuts. You will find a wonderful exposition to Dedekind cuts in Walter Rudin Principles of Mathematical Analysis.

A yet another axiomatic way to think about the real numbers is that it is a complete ordered field, the natural numbers is the smallest inductive subset of real numbers, the integers are the subroup generated by natural numbers, and the rationals are the field of fraction of integers.

Pick and choose which you like. Any further discussion is welcome.

Solution 2:

We have this Platonic ideal in mind when we think of the real numbers: Numbers on a line, densely ordered like the rationals, and like a line it doesn't have "any holes", i.e. some kind of completeness property (see the article Real Analysis in Reverse by James Propp).

It's easy to show that any two ordered fields, $R$ and $R'$, with the least upper bound property, say, are uniquely isomorphic, i.e. there exists a unique order-preserving field isomorphism $\varphi: R\to R'$. This guarantees that if we only use that the reals are an ordered field with the least upper bound property, then any statement we make using only those facts will be true for any set representation that has those properties as well.

In short: It doesn't really matter. All that matters is that there is some set representation and you're free to pick whichever one you like the best. I'm personally partial to the "equivalence classes of rational Cauchy sequences" and I've outlined a proof of existence and uniqueness in another answer of mine.