"Real numbers make possible limit operations with rational numbers, but they would be of little value if the corresponding limit operations carried out with them necessitated the introduction of some further kind of "unreal" numbers which would have to be fitted in between the real ones, and so on ad infinitum. Fortunately, the definition of real number is so comprehensive that no further extension of the number system is possible without discarding one of its essential properties." - Introduction to Calculus and Analysis: Volume I by Richard Courant and Fritz John
Author's Commentary (Hong Kong China, 03/15/2026): It really has been hectic... although I hope it can become even more hectic in the future, besides, my experience continues to grow. There are approximately 112 hours of waking hours per week, taking away 40 hours is nothing, 80 hours still leaves significant room.
0 Introduction
Suppose we were provided a problem of trying to discover an optimal shape for the wheels of a vehicle traveling over terrain that is expected to be reasonably flat in an average sense.$^{[1]}$ Then, possibly from empirically informed intuition, we may suspect that the circle is most optimal in the sense that, for instance, every point on the circumference of a circle is equidistant from its center and so there may be a "continuous falling" phenomenon as a wheel rolls along reasonably flat surfaces.$^{[2]}$ Yet, there is no such thing as a "perfect circle", or, at the very least, according to the limits of modern physics as far as I am aware, there cannot be a perfect circle due to various phenomena as in the phenomenon of Heisenberg's uncertainty principle. So, at best, we would merely provide polygons, perhaps smoother and smoother, which approximate the circle in its abstract conception.
Then, a natural question arises. Could generalized approximation processes, algorithms, be provided that allow one to approximate the circle with greater and greater precision? In fact, can even more general conceptions be realized so as to extend beyond that of the circle? As a partial answer, one of the successful objects as utilized in the history of mathematics comes in the Cauchy sequences, and, in fact, equivalence classes$^{[3]}$ of Cauchy sequences have demonstrated their tremendous utilitarianism in certain more modern topics of analysis.
1 Historical Results in Approximations to Real Numbers
Recall that Cauchy sequences are sequences $\{ x_n \}_{n = 1}^{\infty}$ such that, for any arbitrarily small real $\epsilon > 0$, one can always find a sufficiently large $N$ such that for $n$ and $m$ greater than $N$, we have that $\left| x_n - x_m \right| < \epsilon$.$^{[4]}$ Intuitively, Cauchy sequences characterize the "squeezing" of convergent sequences without a priori knowledge of the limits of sequences. Regarding certain historical results in the appraisal of real numbers, it was always thought that, at least by certain mathematicians and scientists, corresponding to each point of the "real line" is a real number,$^{[5]}$ and that such real numbers can of course be appraised via sequences of rational numbers, or, more appropriately, appropriate Cauchy sequences of rational numbers, especially in light of the fact that irrational numbers can never be expressed completely if one were to, say, utilize the decimal number system in the sense that irrational numbers, in their decimal expansions, have periods that are infinite.
A natural approach then as taken by many mathematicians was of course the intuitive approach that allows the iterative intuitive appraisals of phenomena on the real line, in the sense of,
Theorem 1.1 (Nested Intervals Theorem) If we were provided an infinite nested sequence of bounded closed intervals with real end-points in the sense of $\{ [a_i, b_i] \}_{i = 1}^{\infty}$ for $[a_m, b_m] \subset [a_n, b_n]$ whenever $n < m$ and such that $| b_i - a_i | \rightarrow 0$ for $i \rightarrow \infty$, then, we can always find some unique real number contained in the intersection of all such closed intervals.$^{[6]}$
PROOF.$^{[7]}$ First we demonstrate that such a real number must clearly be unique. If such a real number is not unique, then there exists, at least, $c$ and $c'$ for $c \neq c'$ in $[a_i, b_i]$ for all $i$ such that $|c - c'| > \epsilon$ for some sufficiently small real $\epsilon > 0$, but then, $b_i - a_i$ would not tend to 0 for $i \rightarrow \infty$, contradicting the provided hypothesis. Now, regarding the existence, we shall provide a proof from a certain conventional axiomatic foundation where the existence of suprema and infima are taken to exist.$^{[8]}$ So, for all $i$, we take the supremum of the set of numbers provided by $\{ a_i \}$ and the infimum of the set of numbers provided by $\{ b_i \}$ as denoted by $A$ and $B$ respectively, and such numbers certainly exist as the non-empty set $\{ a_i \}$ is bounded above by $b_1$, and, the non-empty set $\{ b_i \}$ is bounded below by $a_1$. Then, since $|b_i - a_i| \rightarrow 0$ for $i \rightarrow \infty$, we have that $A = B$ and, in fact, such a real number is the real number that is recovered in the provided intersection of the infinite nested sequence of bounded closed intervals with real end-points. $\square$
We comment, with the use of infinite nested sequences of bounded closed intervals with rational end-points rather than real end-points, it is actually possible to simply define real numbers in general, and, in particular, irrational numbers, by appropriate equivalence classes of nested intervals. If, in the limit, an infinite nested sequence of bounded closed intervals with rational end-points was to recover a rational number, then the rational number is provided. However, if, in the limit, no rational numbers are recovered, then we take the provided infinite sequence of nested intervals as, for all intents and purposes, representing a real number in general - that is, real numbers can be represented by their approximations as realized with infinite nested sequences of bounded closed intervals with rational end-points. Before we proceed to the next theorem, we remind the reader of the notion of "metric completion". A "space" is completed with respect to a metric if every Cauchy sequence converges to some object of the space in the metric. Such a definition is of course informal and various important subtleties have been missed, and we intend to touch on them very quickly before moving on, so consider two subtleties in,
I. Cauchy sequences are realized with the metrics defined. What is a completed space with respect to some metric may no longer be completed with respect to another metric.
II. In completed spaces, every Cauchy sequence is to converge to objects "of the same class", "of the same space". For example, there exist Cauchy sequences of rational numbers that do not converge to rational numbers, then, since there exists Cauchy sequences of rational numbers that do not converge to objects of the same class, the set, or class, of rational numbers, is not a completed set with respect to the conventional Archimedean metric, being, the "distance metric" of the real line.
Theorem 1.2 The real line $\mathbb{R}$ is a complete metric space with respect to the conventional Archimedean metric.$^{[9]}$
PROOF. Since every Cauchy sequence of real numbers allows one to provide nested sequences of closed intervals, we have then that every Cauchy sequence of real numbers converges to a real number and thus the completion. To see this, if we were given a Cauchy sequence, then, for any arbitrarily small real $\epsilon > 0$, one can always find some sufficiently large $N$ such that $|x_n - x_m| < \epsilon$ for $n$ and $m$ greater than $N$, that is, such a process could be iterated to provide a nested sequence of closed intervals. Thus, suppose we were given some $\epsilon_1$, then we find the smallest $N_1$ such that $|x_n - x_m| < \epsilon_1$ for all $n$ and $m$ greater than $N_1$. Then, we continue to $\epsilon_2 < \epsilon_1$, and we now find the smallest $N_2$ such that $|x_n - x_m| < \epsilon_1$ and $|x_n - x_m| < \epsilon_2$ for $n$ and $m$ greater than $N_2$. In general, for $\epsilon_{k} < \epsilon_{k - 1} < \dots < \epsilon_{1}$, we find the smallest $N_{k}$ such that $\left( |x_n - x_m| < \epsilon_1 \right) \land \dots \land \left( |x_n - x_m| < \epsilon_k \right)$ for $n$ and $m$ greater than $N_k$. We continue in such a fashion indefinitely, producing nested sequences of closed intervals with lengths $\epsilon_1 > \epsilon_2 > \dots$ for $N_1 < N_2 < \dots$, that is, with $\epsilon \rightarrow 0$ for $N \rightarrow \infty$. And so, an application of Theorem 1.1 completes the argument. $\square$
We emphasize again that real numbers in general can be characterized abstractly in the sense of their approximation schemes. Next, consider a theorem that existed even during the Middle Ages,
Theorem 1.3 Any real number can be represented decimally.
PROOF. In recalling that real numbers in general can be represented by approximation schemes that make use of rational numbers only, all that is required is to demonstrate that any rational number can be represented decimally, and that such a set of objects is dense in the set of reals. If so, then equivalence classes of Cauchy sequences are all that are required to realize the set of real numbers via decimal numbers. Indeed, without losing generality, we recall that a rational number is given by a ratio of integers $\frac{a}{b}$, and all one needs to do is to apply the Euclidean algorithm decimally in order to express such a rational number decimally. Then, an application of a variation of Theorem 1.1 completes the proof, where we use nested sequences of closed intervals with decimally represented rational end-points rather than real end-points.$^{[10]}$ $\square$
So, ultimately, in light of Theorem 1.1, Theorem 1.2, and Theorem 1.3, we can actually ascertain as a result that the set of real numbers actually suffices as a complete system of measurement in various contexts including in contexts of constructions of general approximation schemes that makes use of relatively simple integers (rational numbers can, of course, be given in terms of integers). Not merely is the set of real numbers completed, but we must repeat yet again an important point, that the set of real numbers can be realized by approximation schemes and thus a complete system of measurement has been constructed which allows the general appraisals of approximation schemes. To be more clear, Theorem 1.1 demonstrates that approximations can be provided always via systemic means that "locate" real numbers with greater and greater precision, Theorem 1.2 demonstrates completeness with respect to the extremely utilitarian Archimedean metric, and Theorem 1.3 demonstrates efficient computational feasibility in the possibility of decimal computations. So, it is unsurprising that real numbers and their decimal representations in particular have proliferated as the dominant form of computational objects.
For an intuitive graphic, consider the following (yes, it will never terminate, not for $\pi$, meaning, for conventional digital computers, it will just overflow),
Footnotes
[1] Average in what sense? For the moment, many simple definitions are possible, as in definitions that may involve integral averages of curvatures of surfaces. But, for the purpose of this blog post, we do not intend to discuss differential geometric phenomena.
[2] If surfaces were not reasonably flat in a reasonable average sense, then other shapes may become optimal. For instance, consider square wheels over certain surfaces with periodic indentations. In fact, for a certain surface consisting of an appropriate kind of periodic indentations, the corners of the square wheels trace out inverted catenaries as they roll along such a surface.
[3] In many contexts, what matters is not so much the particular Cauchy sequences themselves, but the equivalence classes of them. In an analogous fashion, in many contexts, it is not the particular representations of rational numbers that are of interest, but the equivalence classes of rational numbers, that, for instance, $\frac{1}{2}$ belongs to the same equivalence class as $\frac{2}{4}$, $\frac{4}{8}$, and so on.
[4] We note that such is not the most general definition of Cauchy sequences. For instance, as is well known, Cauchy sequences could be defined with the more general notion of a metric.
[5] Consider the various illustrations of approximations of real numbers via sequences of rational numbers with the use of continued fractions, or infinite series, in such historical documents as Newton's The Method of Fluxions and Infinite Series: Applications to the Geometry of Curve-Lines.
[6] We require that the intervals be closed, since, if the intervals were open, then it may just so happen that the infinite intersection of all such open intervals produces the empty set. Incidentally, such a result is one of the classic results of analysis that allows one to recall, for instance, that countable intersections of closed sets produce closed sets, where, in such a context of nested intervals, we have that the set as produced is a closed singleton set.
[7] In truth, many such kinds of "proofs" can be somewhat ambiguous if the axiomatic system is not agreed upon. Indeed, in our provided proof, notice that, by supposing that one can always find some sufficiently small real $\epsilon$ such that $|c - c'| > \epsilon$ for $c \neq c'$, one is assuming that there is no such thing as a "number" that is smaller than any arbitrarily small real number, in the sense that there does not exist some kind of an "infinitesimal" $\epsilon$ such that $c + \epsilon = c$. Incidentally, such assumptions had provided the axiomatic foundations of an early style of infinitesimal calculus, a style that we do not adopt. For more information, please see a fascinating annotated historical exposition in L'Hôpital's Analyse des infiniments petits: An Annotated Translation with Source Material by Johann Bernoulli - indeed, compare with Courant's quote as cited at the beginning of this blog post. Additionally, many subtleties were not explicitly appraised, as in the subtlety of the fact that such sequences of nested intervals are obviously indexed by the countable set of natural numbers - recall that, for instance, according to the axioms of general topology, an arbitrary intersection of closed sets produces a closed set.
[8] Actually, a conventional axiomatic system in analysis is the ZFC set-theoretic foundation. But, for the purpose of our proof, the Peano axioms will even suffice - although the Peano axioms do not engage with certain subtleties of general topology and the uncountable, our proof makes the transition to the countably infinite rather than the uncountably infinite. For slightly more clarity, we may sketch some suggestions here. The Peano axioms provide the natural numbers, then, with the introduction of an additional "minus" arithmetic map $-$, we transition to the integers as appropriate equivalence classes of natural numbers. After, we provide rational numbers via appropriate equivalence classes of integers, and, transition to the real numbers via appropriate equivalence classes of rational numbers, where the existence of suprema is realized - indeed, a popular approach is to realize equivalence classes of rational numbers in using "Dedekind cuts" with no explicit references to Cauchy sequences.
[9] In identifying such a notion as an "Archimedean metric", we acknowledge that there may exist non-Achimedean metrics where attempts of metric completions will produce different kinds of mathematical spaces. An example is of course provided by the set of p-adics, although, according to Ostrowski's theorem, up to an isomorphism, the set of p-adics is the only alternative to the set of reals as a metric completion of the set of rational numbers.
[10] Despite the feigned simplicity of the supposed "proof", many subtleties have been missed, culminating in millenniums of mathematical developments in such topics as arithmetic, algebra, the theory of continued fractions, and so on. For instance, Joseph's The Crest of the Peacock: The Non-European Roots of Mathematics documents various algorithms of iterated decompositions of numbers that had existed during the Middle Ages.
References
Bogachev, V. I., & Smolyanov, O. G. (2020). Real and Functional Analysis. Springer.
Courant, R., & Fritz, J. (1989). Introduction to Calculus and Analysis: Volume I. Springer New York. (Original work published 1965)
Stein, E. M., & Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press.
Stein, E. M., & Shakarchi, R. (2005). Real Analysis: Measure Theory, Hilbert Spaces, & Integration. Princeton University Press.
Zorich, V. A. (2016). Mathematical Analysis I (2nd ed.). Springer Berlin.
Zorich, V. A. (2016). Mathematical Analysis II (2nd ed.). Springer Berlin.
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