02.02.02 · analysis / real-numbers

The real numbers — Dedekind cuts, Cauchy completeness, and uniqueness

shipped3 tiersLean: none

Anchor (Master): Rudin, Principles Ch. 1; Royden, Real Analysis Ch. 1–2; Dieudonné, Foundations of Modern Analysis Ch. 2

Intuition Beginner

The rational numbers are the fractions — numbers like $0$ , $1$ , $- 7$ , $\frac{1}{2}$ , $\frac{22}{7}$ . They sit densely on the line: between any two fractions there is always another fraction. Yet the rational line has holes. The length of the diagonal of a unit square, namely $2$ , is not a fraction, and no fraction squares to exactly $2$ . The real numbers are what you get by filling in every such hole so that the line is genuinely continuous.

There are two classical recipes for the filling, and this unit builds both. Richard Dedekind's recipe is to cut the rationals into a lower piece and an upper piece; each cut is one real number. Georg Cantor's recipe is to nail a real number down by a Cauchy sequence of rationals — a list that closes in on its target. The two recipes produce the same line, and the reason they agree is that there is only one complete ordered number line up to renaming.

The companion unit 02.02.01 lists the thirteen rules the real numbers satisfy and takes those rules as given. This unit does the opposite: it builds a structure from the rationals and then checks, rule by rule, that the result satisfies those thirteen axioms. Building the line is what proves the axioms are consistent — that some object actually obeys them.

Visual Beginner

Picture the rational line as a row of dots — dense, yet scattered with gaps. A Dedekind cut slides a vertical marker across the line: every rational to the left of the marker lands in the lower piece, every rational to the right lands in the upper piece. When the marker sits at an irrational spot like $2$ , the lower piece is a cut that names a point no single rational can name.

The lower piece is "everything below the marker." The marker itself is the real number the cut represents — even when no rational sits exactly at the marker.

Worked example Beginner

Build the real number $2$ as a Dedekind cut. Split the rationals: put a rational $q$ into the lower piece whenever $q < 0$ or $q^{2} < 2$ , and leave the remaining rationals above. Call this lower piece $α$ .

Which rationals land in $α$ ? The number $- 3$ is in, because it is negative. So are $0$ , $1$ , and $1.4$ , since $1. 4^{2} = 1.96 < 2$ . The rational $1.41$ is in ( $1.4 1^{2} = 1.9881 < 2$ ), and so is $1.414$ ( $1.41 4^{2} = 1.999396 < 2$ ). But $1.42$ is out ( $1.4 2^{2} = 2.0164 > 2$ ), and so are $1.415$ and $2$ .

The lower piece $α$ has no largest member: for every rational inside it, a slightly bigger rational is still inside. Its upper edge sits exactly at $2$ , the number whose square is $2$ . The cut is the way you point at $2$ using nothing but rationals — you never name $2$ directly, you name the set of rationals that lie below it.

Check your understanding Beginner

Formal definition Intermediate+

This section gives the two constructions in parallel: the cut reals $R_{D}$ and the Cauchy reals $R_{C}$ . The rationals $Q$ and their order $<$ are taken as given.

Definition (Dedekind cut). A Dedekind cut is a subset $α \subseteq Q$ satisfying three conditions:

(proper) $α \neq = \emptyset$ and $α \neq = Q$ ;
(downward closure) if $p \in α$ and $q \in Q$ satisfies $q < p$ , then $q \in α$ ;
(no maximum) for every $p \in α$ there exists $r \in α$ with $p < r$ .

Write $R_{D}$ for the set of all cuts. A rational $q \in Q$ embeds as the principal cut $q^{*} := {p \in Q : p < q}$ . Order cuts by inclusion: $α \leq β ⟺ α \subseteq β$ . Addition is $α + β := {a + b : a \in α, b \in β}$ . Multiplication is defined first on positive cuts by $α \cdot β := {a \cdot b : a \in α, a > 0, b \in β, b > 0} \cup {q \in Q : q \leq 0}$ , then extended to all signs by sign cases ^{[Rudin Ch. 1]}.

Definition (Cauchy sequence and the Cauchy reals). A sequence $(a_{n})_{n \geq 1}$ in $Q$ is Cauchy if for every $ε > 0$ in $Q$ there exists $N \in N$ such that $∣ a_{m} - a_{n} ∣ < ε$ for all $m, n \geq N$ . Two Cauchy sequences are *equivalent*, written $(a_{n}) \sim (b_{n})$ , if $∣ a_{n} - b_{n} ∣ \to 0$ in $Q$ , meaning for every $ε > 0$ some $N$ has $∣ a_{n} - b_{n} ∣ < ε$ for all $n \geq N$ .

The Cauchy reals are the quotient $R_{C} := C / \sim$ , where $C$ is the set of rational Cauchy sequences. Write $[(a_{n})]$ for the equivalence class of $(a_{n})$ . Operations are termwise: $[(a_{n})] + [(b_{n})] = [(a_{n} + b_{n})]$ and $[(a_{n})] \cdot [(b_{n})] = [(a_{n} \cdot b_{n})]$ . The rational $q$ embeds as the constant sequence $[(q, q, q, \dots)]$ . Order: $[(a_{n})] < [(b_{n})]$ iff some $ε > 0$ in $Q$ and some $N \in N$ satisfy $b_{n} - a_{n} \geq ε$ for all $n \geq N$ .

Why two constructions? The cut model makes the order and least-upper-bound structure immediate (a cut is a lower set). The Cauchy model makes the metric and convergence structure immediate (a real is a converging sequence). Each construction makes some axioms of 02.02.01 immediate and forces the others through a verification. That the two quotients agree is the content of the uniqueness theorem below.

Key theorem with proof Intermediate+

Theorem (completeness of the cut construction). Let $A \subseteq R_{D}$ be a non-empty set of cuts that is bounded above. Then $sup A$ exists in $R_{D}$ , and

sup A = α \in A ⋃ α .

In words: the supremum of a bounded family of cuts is the union of those cuts.

Proof. Set $β := ⋃_{α \in A} α \subseteq Q$ . We verify the three cut axioms, then that $β$ is the least upper bound of $A$ .

(Proper.) Because $A$ is non-empty, some $α_{0} \in A$ is non-empty, so $β \supseteq α_{0} \neq = \emptyset$ . Because $A$ is bounded above, some cut $γ$ satisfies $α \subseteq γ$ for every $α \in A$ , hence $β \subseteq γ ⊊ Q$ ; so $β \neq = Q$ .

(Downward closure.) Take $p \in β$ and $q < p$ in $Q$ . Then $p \in α_{0}$ for some $α_{0} \in A$ , and downward closure of $α_{0}$ gives $q \in α_{0} \subseteq β$ .

(No maximum.) Take $p \in β$ . Then $p \in α_{0}$ for some $α_{0} \in A$ . Since $α_{0}$ has no maximum, some $r \in α_{0}$ has $r > p$ . Because $α_{0} \subseteq β$ , this $r$ lies in $β$ with $r > p$ .

So $β$ is a cut. It is an upper bound of $A$ because $α \subseteq β$ for every $α \in A$ , i.e. $α \leq β$ . It is the least upper bound: if $γ$ is any upper bound of $A$ , then $α \subseteq γ$ for every $α \in A$ , so $β = ⋃_{α} α \subseteq γ$ , i.e. $β \leq γ$ . Therefore $β = sup A$ . $□$

The parallel statement holds for $R_{C}$ : a bounded non-empty set of Cauchy reals has a supremum, proved by selecting representative sequences and tracking their eventual behaviour. Each construction delivers the completeness axiom (C) of 02.02.01 as a theorem rather than a postulate.

Bridge. The union-of-cuts formula is the foundational reason the cut model satisfies completeness — the supremum is not introduced by fiat but read off directly from the lower-set data, and this is exactly the moment the construction earns the right to be called a model of the real numbers. The pattern "supremum as a closure operation on lower sets" builds toward the order-theoretic view of completeness in lattice theory and domain theory, and appears again in the metric-space completion of 02.01.05 (where the Cauchy-completion half of this unit generalises from $Q$ to any metric space), in the sequence convergence theory of 02.03.02, and in the measure-theoretic constructions of 02.07.02. Putting these together, the bridge is the recognition that every rigorous construction of $R$ is a different route to the same completeness property, and that property is what the rest of analysis repeatedly exploits.

Exercises Intermediate+

Exercise 5 (medium, proof).

Show that the termwise sum of two rational Cauchy sequences is again Cauchy, and conclude that addition on $R_{C}$ is well-defined (independent of the chosen representatives).

Hint

Use $∣ (a_{m} + b_{m}) - (a_{n} + b_{n}) ∣ \leq ∣ a_{m} - a_{n} ∣ + ∣ b_{m} - b_{n} ∣$ for Cauchyness, and the triangle inequality on $∣ (a_{n} + b_{n}) - (a_{n}^{'} + b_{n}^{'}) ∣$ for well-definedness.

Answer

Cauchyness: given $ε > 0$ , pick $N$ so that $∣ a_{m} - a_{n} ∣ < ε /2$ and $∣ b_{m} - b_{n} ∣ < ε /2$ for $m, n \geq N$ ; then $∣ (a_{m} + b_{m}) - (a_{n} + b_{n}) ∣ \leq ∣ a_{m} - a_{n} ∣ + ∣ b_{m} - b_{n} ∣ < ε$ . Well-defined: if $(a_{n}) \sim (a_{n}^{'})$ and $(b_{n}) \sim (b_{n}^{'})$ , then $∣ (a_{n} + b_{n}) - (a_{n}^{'} + b_{n}^{'}) ∣ \leq ∣ a_{n} - a_{n}^{'} ∣ + ∣ b_{n} - b_{n}^{'} ∣ \to 0$ , so $(a_{n} + b_{n}) \sim (a_{n}^{'} + b_{n}^{'})$ and $[(a_{n})] + [(b_{n})] = [(a_{n}^{'})] + [(b_{n}^{'})]$ . Multiplication is handled the same way (after first bounding a Cauchy sequence), so all operations on $R_{C}$ are well-defined.

Exercise 6 (hard, proof).

Prove the trichotomy for $R_{C}$ : for every $[(a_{n})] \in R_{C}$ , exactly one of $[(a_{n})] < 0$ , $[(a_{n})] = 0$ , $[(a_{n})] > 0$ holds.

Hint

If $[(a_{n})] \neq = 0$ , then $∣ a_{n} ∣$ does not tend to $0$ ; combine this with the Cauchy property to show $∣ a_{n} ∣$ is eventually bounded below by a positive $ε$ , then rule out sign oscillation.

Answer

Suppose $[(a_{n})] \neq = 0$ , so $∣ a_{n} ∣ \neq \to 0$ : there is $ε > 0$ such that for every $N$ some $n \geq N$ has $∣ a_{n} ∣ \geq ε$ . Because $(a_{n})$ is Cauchy, choose $N_{0}$ with $∣ a_{m} - a_{n} ∣ < ε /2$ for $m, n \geq N_{0}$ . Fix $n_{0} \geq N_{0}$ with $∣ a_{n_{0}} ∣ \geq ε$ . Then for all $n \geq N_{0}$ , $∣ a_{n} ∣ \geq ∣ a_{n_{0}} ∣ - ∣ a_{n} - a_{n_{0}} ∣ > ε - ε /2 = ε /2$ . In particular consecutive terms beyond $N_{0}$ have the same sign (their difference is $< ε /2 < ε /2 + ε /2 = ∣ a_{n} ∣$ , so they cannot straddle zero). Hence either $a_{n} > ε /2$ for all $n \geq N_{0}$ (giving $[(a_{n})] > 0$ ) or $a_{n} < - ε /2$ for all $n \geq N_{0}$ (giving $[(a_{n})] < 0$ ). At most one case holds by the order definition, so exactly one of the three holds.

Exercise 7 (hard, proof).

Construct an explicit order-preserving field homomorphism $φ : R_{D} \to R_{C}$ by sending each cut $α$ to a Cauchy sequence of rationals in $α$ that converges to the boundary of $α$ .

Hint

Use bisection: maintain rationals $ℓ_{n} \in α$ and $u_{n} \in / α$ with $u_{n} - ℓ_{n} = (u_{0} - ℓ_{0}) / 2^{n}$ , and set $φ (α) = [(ℓ_{n})]$ .

Answer

Pick $ℓ_{0} \in α$ and $u_{0} \in / α$ (both exist because $α$ is a proper, non-empty downward-closed set). Recursively define $m_{n} = (ℓ_{n} + u_{n}) /2$ : if $m_{n} \in α$ set $ℓ_{n + 1} = m_{n}$ , $u_{n + 1} = u_{n}$ ; otherwise set $ℓ_{n + 1} = ℓ_{n}$ , $u_{n + 1} = m_{n}$ . Then $ℓ_{n} \in α$ , $u_{n} \in / α$ , and $u_{n} - ℓ_{n} = (u_{0} - ℓ_{0}) / 2^{n}$ , so $(ℓ_{n})$ is Cauchy. Define $φ (α) = [(ℓ_{n})] \in R_{C}$ . For a principal cut $q^{*}$ , the algorithm recovers the constant sequence at $q$ , so $φ (q^{*}) = [(q, q, \dots)]$ and $φ$ fixes the rationals. Order preservation holds: if $α ⊊ β$ , pick $q \in β ∖ α$ ; eventually the bisection for $α$ stays below $q$ while that for $β$ reaches $q$ , giving $φ (α) < φ (β)$ . Preservation of $+$ and $\cdot$ follows from the corresponding recursions on sums and products.

Exercise 8 (hard, short-answer).

Explain why the map $φ$ of Exercise 7 is an isomorphism $R_{D} ≅ R_{C}$ , and state which theorem from 02.02.01 does the heavy lifting.

Hint

Both $R_{D}$ and $R_{C}$ are complete ordered fields. Recall the categoricity theorem.

Answer

By the Key theorem above, $R_{D}$ has the least-upper-bound property and is therefore a complete ordered field; a parallel argument shows $R_{C}$ is a complete ordered field as well. The categoricity theorem of 02.02.01 (any two complete ordered fields are uniquely order-isomorphic) then guarantees that $φ : R_{D} \to R_{C}$ is a bijective order-preserving field isomorphism. Hence the two constructions produce "the same" real numbers, and the choice between cuts and Cauchy sequences is mathematically inert — a matter of convenience, not of content.

Lean formalization Intermediate+

lean_status: none — Mathlib's Real is built once, as Cauchy-completion of $Q$ , and equipped with the LinearOrderedField, ConditionallyCompleteLinearOrderedField, and Archimedean instances. The present unit's pedagogical point — a second model (Dedekind cuts) constructed from scratch, verified axiom-by-axiom, and then proved isomorphic to the first — has no direct Mathlib counterpart: there is no named DedekindCut type carrying its own completeness proof, and no single named isomorphism between two competing models. the Mathlib gap analysis records this gap in full. A learner who wants to formalise the union-of-cuts supremum proof would build a small standalone structure rather than reuse Real.

Advanced results Master

The two constructions of $R$ are not merely two roads to the same object: their agreement is a theorem, and the theorem is categoricity. This section gathers the equivalence, its consequences, and the cardinality fact that separates $R$ from $Q$ .

Theorem (equivalence of the two constructions). The cut reals $R_{D}$ and the Cauchy reals $R_{C}$ are each complete ordered fields, and the map $φ : R_{D} \to R_{C}$ of Exercise 7 is a unique order-preserving field isomorphism.

The completeness of $R_{D}$ is the Key theorem (the supremum is a union of cuts). The completeness of $R_{C}$ is proved by selecting representative sequences for a bounded set and constructing a limiting sequence from their eventual values. Categoricity — proved in 02.02.01 and reproved constructively in the Full proof set below — then promotes the homomorphism $φ$ to an isomorphism.

Theorem (Archimedean property, constructive form). In either model, for every real $x$ there is an integer $n$ with $x < n$ .

In the cut model the proof is one line: a cut $α$ is not all of $Q$ , so some rational $r \in / α$ bounds it from above, and any integer $n > r$ satisfies $α \subseteq n^{*}$ . This is the same Archimedean fact stated axiomatically in 02.02.01, now derived from the construction. The density of $Q$ in $R$ follows by the same Archimedean-denominator argument.

Theorem (Cantor — $R$ is uncountable). There is no surjection $N \to R$ ; equivalently, every sequence of reals misses some real.

The proof given in the Full proof set uses decimal expansions and a diagonal swap. The companion unit 02.02.01 gives the nested-interval version; the two are equivalent, but the decimal diagonal makes the mechanism — "change the $n$ -th digit of the $n$ -th number" — maximally concrete. The rationals, by contrast, are countable (list fractions by increasing height $∣ p ∣ + q$ ), so the diagonal argument separates $R$ from $Q$ on cardinality grounds: there are strictly more reals than rationals.

Remark (constructive vs classical). In Bishop-style constructive analysis the Cauchy-completion model is preferred because it is computable: a real number is a Cauchy sequence together with an explicit modulus of convergence. The least-upper-bound property, however, is not constructively valid for arbitrary inhabited bounded sets (it implies a form of the limited principle of omniscience). So in the constructive setting the two models pull apart: Cauchy reals remain well-behaved, while the full supremum operation is a strictly classical principle. This refines the classical equivalence: classically the models agree, constructively they diverge exactly on the LUB axiom.

Synthesis. Putting these together, the three results of this section — equivalence of the two constructions, the Archimedean property derived rather than postulated, and Cantor uncountability — exhibit the real line as a single object characterised up to unique isomorphism by completeness, order, and field structure. The central insight is that the construction's job is to prove consistency (a model exists) and that categoricity's job is to prove uniqueness (the model is the only one); together they license the textbook practice of speaking of the real numbers without specifying a construction. The foundational reason the two constructions agree is that both satisfy the same complete-ordered-field axioms, and the equivalence generalises to metric-space completions 02.01.05, where any metric space embeds densely in a unique complete one. This builds toward the $L^{p}$ -completions of 02.07.06 and the Sobolev-space completions of [02.16.x], and the sup-and-sequence machinery appears again in the integration theory of 02.04.04 and the measure theory of 02.07.02, where limits of sequences of functions replace limits of sequences of rationals.

Full proof set Master

Proposition ( $R_{D}$ is Archimedean). For every cut $α \in R_{D}$ there exists $n \in N$ with $α < n^{*}$ .

Proof. Because $α$ is a cut, $α \neq = Q$ , so pick $r \in Q ∖ α$ . For any $a \in α$ , the inequality $a \geq r$ is impossible: if $a > r$ then downward closure of $α$ forces $r \in α$ , a contradiction, and if $a = r$ then $r \in α$ , also a contradiction. Hence $a < r$ for every $a \in α$ , and $α \subseteq {p \in Q : p < r}$ . Choose an integer $n > r$ (integers are unbounded in $Q$ ). Then ${p : p < r} \subseteq {p : p < n} = n^{*}$ , so $α \subseteq n^{*}$ ; and $r \in n^{*} ∖ α$ (since $r < n$ but $r \in / α$ ), so $α ⊊ n^{*}$ , i.e. $α < n^{*}$ . $□$

Proposition (density of $Q$ in $R_{D}$ ). For cuts $α < β$ there exists $q \in Q$ with $α < q^{*} < β$ .

Proof. Since $α ⊊ β$ , pick $s \in β ∖ α$ . Because $β$ has no maximum, pick $r \in β$ with $r > s$ . Then $r \in / α$ (otherwise $s < r$ and downward closure of $α$ would give $s \in α$ ). We claim $α ⊊ r^{*} ⊊ β$ . For the first inclusion: any $a \in α$ satisfies $a < r$ (else $a \geq r$ forces $r \in α$ by downward closure, contradiction), so $a \in r^{*}$ , giving $α \subseteq r^{*}$ ; and $s < r$ with $s \in / α$ gives $s \in r^{*} ∖ α$ , so the inclusion is strict. For the second: $r \in β$ and $β$ is downward closed, so $r^{*} = {p : p < r} \subseteq β$ ; and $β$ has no maximum, so some $t \in β$ has $t > r$ , giving $t \in β ∖ r^{*}$ , so the inclusion is strict. Setting $q = r$ yields $α < q^{*} < β$ . $□$

Proposition (Cantor's diagonal theorem — $R$ is uncountable). The open interval $(0, 1)$ is uncountable: no sequence $(x_{n})_{n \geq 1}$ in $(0, 1)$ lists every real of $(0, 1)$ .

Proof. Suppose, towards a contradiction, that $(x_{n})$ is a sequence in $(0, 1)$ . Write each $x_{n}$ in decimal form $x_{n} = 0. d_{n 1} d_{n 2} d_{n 3} \dots$ with each $d_{nk} \in {0, 1, \dots, 9}$ , choosing the non-terminating expansion whenever a number has two (so $0.5000 \dots$ is written as $0.4999 \dots$ ). Construct a new decimal $y = 0. e_{1} e_{2} e_{3} \dots$ digit by digit, setting

e_{n} = {23 if d_{nn} \neq = 2, if d_{nn} = 2.

Every digit $e_{n}$ lies in ${2, 3}$ , so the decimal $y$ has a unique expansion (it has no trailing $9$ s and no trailing $0$ s) and satisfies $0 < y < 1$ . By construction $e_{n} \neq = d_{nn}$ for every $n$ , so $y$ differs from $x_{n}$ in the $n$ -th digit, hence $y \neq = x_{n}$ for every $n$ . Therefore $y \in (0, 1)$ is not in the list, contradicting the assumption that $(x_{n})$ exhausts $(0, 1)$ . So no such listing exists, and $(0, 1)$ — hence $R$ — is uncountable. $□$

Proposition (categoricity, constructive proof sketch). Let $F_{1}, F_{2}$ be two complete ordered fields. There is a unique order-preserving field isomorphism $ψ : F_{1} \to F_{2}$ .

Proof sketch. Define $ψ$ on the natural numbers of $F_{1}$ by $ψ (1) = 1$ and $ψ (n + 1) = ψ (n) + 1$ , extend to $Z$ by $ψ (- n) = - ψ (n)$ , and to $Q$ by $ψ (m / n) = ψ (m) \cdot ψ (n)^{- 1}$ . For an arbitrary $x \in F_{1}$ , the lower set $L_{x} = {q \in Q : q < x}$ is non-empty and bounded above (by Archimedean), so define $ψ (x) = sup_{F_{2}} ψ (L_{x})$ , which exists by completeness of $F_{2}$ . One checks $ψ$ preserves $+$ , $\cdot$ , $<$ by approximating through rationals and using the supremum characterisation. Uniqueness holds because any order-preserving field isomorphism fixes $Q$ , and the lower set $L_{x}$ determines each $x$ via the supremum. Applied to $F_{1} = R_{D}$ and $F_{2} = R_{C}$ , this recovers the isomorphism of Exercise 8. $□$

Connections Master

Real-number axioms 02.02.01. The companion unit postulates the thirteen axioms and derives their consequences; this unit constructs a model and checks the axioms hold. The two units are complementary — 02.02.01 gives the structural fingerprint, this unit proves the fingerprint is consistent and categorical. The categoricity theorem stated there is reproved constructively in the Full proof set above.
Topological space 02.01.01 and metric space 02.01.05. The Cauchy-completion half of this unit is the special case of the metric-space completion of 02.01.05 applied to $(Q, ∣ \cdot ∣)$ ; the topology that the resulting $R$ carries is the metric topology, a concrete instance of the abstract topological-space framework of 02.01.01. Generalising from $Q$ to an arbitrary metric space yields the Banach-space completions of 02.11.04.
Sequences, Cauchy completeness, and Bolzano-Weierstrass 02.03.02. The Cauchy reals are built from Cauchy sequences of rationals; the general $ε$ - $N$ theory of sequence convergence, Bolzano-Weierstrass, and the equivalence of Cauchy completeness with the least-upper-bound property are developed in 02.03.02. This unit supplies the motivating example (the completion of $Q$ ); that unit supplies the general theory.
Fundamental theorems of calculus 02.04.04. The existence of the Darboux/Riemann integral of a continuous function on a compact interval, and the proof of the fundamental theorem, rely on the least-upper-bound property at every step — via the extreme value theorem, uniform continuity, and the sup/inf of approximation sums. This unit shows where that property comes from.
Measure theory and Lebesgue measure 02.07.01, 02.07.02. The Borel $σ$ -algebra on $R$ is generated by the open intervals of the topology this unit constructs, and Lebesgue outer measure is defined on the power set of the $R$ built here. The very definition of a measurable set presupposes a completed real line; this unit is its construction.

Historical & philosophical context Master

The problem of giving the real numbers a rigorous foundation — independent of geometric intuition about "the line" — drove nineteenth-century analysis. Richard Dedekind's Stetigkeit und irrationale Zahlen (1872) introduced the cut construction after a long incubation: Dedekind recorded in the preface that the idea came to him on the morning of 24 November 1858, while teaching the foundations of the integral in Zürich ^{[Dedekind 1872]}. Dedekind's motivation was pedagogical as much as research-driven: he found he could not prove, without circularity, that a monotonically increasing bounded sequence converges, and the cut was his remedy.

Georg Cantor, working independently on the uniqueness of trigonometric series representations, arrived at the Cauchy-completion construction in the same year, publishing "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen" (1872) ^{[Cantor 1872]}. Cantor's "number domains" (Zahlengrössen) are precisely the equivalence classes of rational Cauchy sequences used in this unit. Karl Weierstrass, in his Berlin lectures through the 1860s and 1870s, developed a parallel arithmetisation of analysis — his construction used convergent series of rationals rather than sequences or cuts — though he published the details chiefly through his students.

The simultaneous discovery by Dedekind, Cantor, and Weierstrass of equivalent constructions is a classical case of a mathematical idea whose time had come: the arithmetisation programme sought to purge analysis of lingering geometric assumptions, and a construction of $R$ from $Q$ was its keystone. David Hilbert's Grundlagen der Geometrie (1899) closed the loop by reversing the strategy: rather than construct $R$ , Hilbert axiomatised it, adding the Vollständigkeitsaxiom (completeness axiom) to the ordered-field axioms and observing that the resulting system is categorical — any two models are isomorphic ^{[Hilbert 1899]}. The two strategies are reconciled in this unit: the constructions prove existence (a model exists, so the axioms are consistent), and categoricity proves uniqueness (the model is essentially the only one). Modern foundational practice, following Bourbaki and Landau, alternates freely between the two — and modern type-theoretic foundations, including Mathlib's Real, almost universally adopt Cantor's Cauchy-completion route.

Bibliography Master

@book{Dedekind1872Stetigkeit,
  author    = {Dedekind, Richard},
  title     = {Stetigkeit und irrationale Zahlen},
  publisher = {Vieweg},
  address   = {Braunschweig},
  year      = {1872}
}

@article{Cantor1872Trigonometrische,
  author  = {Cantor, Georg},
  title   = {Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen},
  journal = {Mathematische Annalen},
  volume  = {5},
  number  = {1},
  pages   = {123--132},
  year    = {1872}
}

@book{Hilbert1899Grundlagen,
  author    = {Hilbert, David},
  title     = {Grundlagen der Geometrie},
  publisher = {Teubner},
  address   = {Leipzig},
  year      = {1899}
}

@book{RudinPrinciples,
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  title     = {Principles of Mathematical Analysis},
  edition   = {3rd},
  publisher = {McGraw-Hill},
  year      = {1976}
}

@book{ApostolCalculusVol1,
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  title     = {Calculus, Volume I: One-Variable Calculus, with an Introduction to Linear Algebra},
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}

@book{RoydenRealAnalysis,
  author    = {Royden, H. L.},
  title     = {Real Analysis},
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  year      = {1988}
}

@book{DieudonneFoundations,
  author    = {Dieudonn{\'e}, Jean},
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  year      = {1960}
}

@book{BourbakiTopologie,
  author    = {Bourbaki, Nicolas},
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  year      = {1971}
}

Prerequisites

02.02.01
02.01.01

Tier anchors

beginner: Intuition: fill the gaps in the rationals — every length on the line is a real number (Strogatz, Infinite Powers, number-line framing)
intermediate: Rudin, Principles of Mathematical Analysis, Ch. 1 (the construction of R); Apostol, Calculus Vol. 1, Ch. 3 §3.1–3.7
master: Rudin, Principles Ch. 1; Royden, Real Analysis Ch. 1–2; Dieudonné, Foundations of Modern Analysis Ch. 2

References

Rudin — Principles of Mathematical Analysis · Ch. 1, the real and complex number systems; ordered-field axioms, least-upper-bound property, and the construction of $R$
Apostol — Calculus Vol. 1 · Ch. 3 §3.1–3.7, field axioms, order axioms, completeness axiom, and verification on the cut model
Dedekind 1872 — Stetigkeit und irrationale Zahlen · the cut construction of $R$ from $Q$ ; Dedekind is the originator of the method
Cantor 1872 — Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen · the Cauchy-completion construction of $R$ as equivalence classes of rational Cauchy sequences
Hilbert 1899 — Grundlagen der Geometrie · §13–§17, the Vollständigkeitsaxiom and the categoricity of the complete ordered field
Royden — Real Analysis · Ch. 1–2, the ordered-field axioms, completeness, Archimedean property, and uniqueness of $R$
Dieudonné — Foundations of Modern Analysis · Ch. 2, real numbers constructed as the completion of $Q$

Estimated time

beginner: 18m
intermediate: 45m
master: 85m