Completeness of the Real Numbers

Contents

Statement
The construction: Dedekind cuts
Order and the least-upper-bound property
The field operations
Consequences

This appendix proves the statement on which all of real and complex analysis rests (Real Analysis): there exists a complete ordered field, unique up to isomorphism — the real numbers. It is the deepest proof of Parts I and II: every convergence theorem of analysis (monotone convergence, Bolzano–Weierstrass, the Cauchy criterion, the intermediate and extreme value theorems, and through them the fundamental theorem of calculus and Cauchy's integral theorem) descends from the single property proven here. The construction is Dedekind's; the rationals $\Q$, with their arithmetic and order, are taken as already constructed (Logic, Sets, and Maps and Algebraic Structures).

Statement

Theorem A7.1 (Completeness of $\R$).

There exists an ordered field $\R$ containing $\Q$ as an ordered subfield, with the least-upper-bound property: every nonempty subset $S \subset \R$ that is bounded above has a least upper bound $\sup S \in \R$. Moreover $\R$ is Archimedean, $\Q$ is dense in it, and every Cauchy sequence in $\R$ converges. Rests on Definition 8.32.

The proof occupies the rest of this section: the construction (The construction: Dedekind cuts), the order and the least-upper-bound property (Order and the least-upper-bound property) — proven before the arithmetic, because it is the property everything else serves — the field operations (The field operations), and the consequences (Consequences).

The construction: Dedekind cuts

The idea is geometric: a real number is identified with the set of all rationals to its left. What must be checked is that the totality of such left rays carries an order, an arithmetic, and no gaps.

Definition A7.2 (Cut).

A cut is a set $\alpha \subset \Q$ such that

$\alpha \neq \varnothing$ and $\alpha \neq \Q$ (nontriviality);
if $p \in \alpha$ and $q < p$ then $q \in \alpha$ (downward closure);
$\alpha$ has no largest element: for every $p \in \alpha$ there is $r \in \alpha$ with $p < r$ (openness).

The set of all cuts is denoted $\R$. Every rational $q$ embeds as the cut $q^{*} = \set{p \in \Q \mid p < q}$; properties (1)–(3) are immediate for $q^{*}$, and $p < q \iff p^{*} \subsetneq q^{*}$, so the embedding preserves order.

Lemma A7.3 (Elementary properties of cuts).

Let $\alpha$ be a cut, $p \in \alpha$, $q \notin \alpha$. Then $p < q$; and every rational $r > q$ satisfies $r \notin \alpha$. Rests on Definition A7.2.

Proof.

Derives Lemma A7.3. If $q \le p$ then downward closure ($q < p$) or membership itself ($q = p$) would put $q \in \alpha$, a contradiction; so $p < q$. If $r > q$ and $r \in \alpha$, downward closure would give $q \in \alpha$ again — contradiction.

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Order and the least-upper-bound property

Definition A7.4 (Order on $\R$).

For cuts $\alpha, \beta$: $\alpha < \beta$ if and only if $\alpha \subsetneq \beta$; and $\alpha \le \beta$ iff $\alpha \subseteq \beta$. Rests on Definition A7.2.

Lemma A7.5 (The order is total).

For any cuts $\alpha, \beta$, exactly one of $\alpha < \beta$, $\alpha = \beta$, $\beta < \alpha$ holds. Rests on Definition A7.4 and Lemma A7.3.

Proof.

Derives Lemma A7.5. At most one holds, since the three are mutually exclusive by construction. Suppose $\alpha \not< \beta$ and $\alpha \neq \beta$; we must show $\beta < \alpha$, i.e. $\beta \subsetneq \alpha$. Since $\alpha \not\subseteq \beta$, pick $p \in \alpha \setminus \beta$. Let $q \in \beta$ be arbitrary. Applying Lemma A7.3 to the cut $\beta$ (with $q \in \beta$, $p \notin \beta$) gives $q < p$; downward closure of $\alpha$ then gives $q \in \alpha$. Hence $\beta \subseteq \alpha$, and the inclusion is strict because $p \notin \beta$.

∎

Theorem A7.6 (Least-upper-bound property).

Let $A \subset \R$ be nonempty and bounded above. Then $\gamma = \bigcup_{\alpha \in A} \alpha$ is a cut, and $\gamma = \sup A$. Rests on Definitions A7.2 and A7.4.

Proof.

Derives Theorem A7.6. $\gamma$ is a cut. Nontriviality: $\gamma \supseteq \alpha_0 \neq \varnothing$ for any $\alpha_0 \in A$; and if $\beta$ is an upper bound of $A$ then every $\alpha \in A$ satisfies $\alpha \subseteq \beta$, so $\gamma \subseteq \beta \subsetneq \Q$. Downward closure: if $p \in \gamma$ then $p \in \alpha$ for some $\alpha \in A$, and any $q < p$ lies in $\alpha \subseteq \gamma$. Openness: the same $\alpha$ contains some $r > p$, and $r \in \gamma$.

$\gamma$ is an upper bound. $\alpha \subseteq \gamma$ for every $\alpha \in A$ by construction, i.e. $\alpha \le \gamma$.

$\gamma$ is the least one. Let $\delta$ be any upper bound of $A$. Then $\alpha \subseteq \delta$ for all $\alpha \in A$, hence $\gamma = \bigcup_{\alpha \in A}\alpha \subseteq \delta$, i.e.\ $\gamma \le \delta$.

∎

Note how little the proof costs once the construction is right: the supremum is literally the union. The gaps of $\Q$ — the cut $\set{p \in \Q \mid p \le 0 \text{ or } p^2 < 2}$ has no rational supremum — are filled by fiat, because the union of cuts is again a cut.

The field operations

Definition A7.7 (Addition).

$\alpha + \beta = \set{p + q \mid p \in \alpha,\ q \in \beta}$, with neutral element $0^{*}$ and $-\alpha = \set{p \in \Q \mid \exists\, r > 0:\ -p - r \notin \alpha}$. Rests on Definition A7.2.

Lemma A7.8 ($(\R, +)$ is an ordered abelian group).

$\alpha + \beta$ and $-\alpha$ are cuts; addition is associative and commutative with $\alpha + 0^{*} = \alpha$ and $\alpha + (-\alpha) = 0^{*}$; and $\alpha \le \beta$ implies $\alpha + \gamma \le \beta + \gamma$. Rests on Definition A7.7, Lemma A7.3 and Definition A7.4.

Proof.

Derives Lemma A7.8. $\alpha+\beta$ is a cut. It is nonempty; it omits $p' + q'$ whenever $p' \notin \alpha$, $q' \notin \beta$ (by Lemma A7.3 any element $p + q < p' + q'$); it is downward closed, since $s < p + q$ writes as $s = (s - q) + q$ with $s - q < p$, so $s - q \in \alpha$; and it has no largest element, since $p$ may be enlarged inside $\alpha$. Associativity and commutativity are inherited from $\Q$ elementwise, and $\alpha + 0^{*} = \alpha$ follows from downward closure and openness: $p + n < p$ for $n < 0$ gives $\alpha + 0^{*} \subseteq \alpha$, while any $p \in \alpha$ has $r \in \alpha$, $r > p$, whence $p = r + (p - r) \in \alpha + 0^{*}$.

Inverse. $-\alpha$ is a cut (checks as above). For $\alpha + (-\alpha) \subseteq 0^{*}$: if $p \in \alpha$ and $q \in -\alpha$ with witness $r > 0$ (so $-q - r \notin \alpha$), then $p < -q - r$ by Lemma A7.3, so $p + q < -r < 0$. Conversely let $s < 0$; put $t = -s/2 > 0$. By the Archimedean property of $\Q$ there is an integer $n$ with $n t \in \alpha$ and $(n+1)t \notin \alpha$ (walk upward in steps of $t$ from inside $\alpha$; the walk exits because $\alpha \neq \Q$). Take $p = n t \in \alpha$ and $q = s - n t$. Then $-q - t = -s + n t - t = n t + 2t - t = (n+1)t \notin \alpha$, so $q \in -\alpha$ with witness $t$, and $p + q = s$. Monotonicity of $+$ is elementwise.

∎

Definition A7.9 (Multiplication).

For $\alpha, \beta > 0^{*}$:

\begin{equation}\tag{A7.1} \alpha\beta = \set{s \in \Q \mid s \le 0} \cup \set{pq \mid p \in \alpha,\ q \in \beta,\ p > 0,\ q > 0}\ec \end{equation}

extended to all signs by $\alpha\, 0^{*} = 0^{*}$ and the sign rules $\alpha\beta = -\left((-\alpha)\beta\right)$ etc., with unit $1^{*}$ and, for $\alpha > 0^{*}$, the reciprocal $\alpha^{-1} = \set{p \in \Q \mid p \le 0} \cup \set{p > 0 \mid \exists\, r > 0:\ (p^{-1} - r)^{-1} \notin \alpha}$. Rests on Definitions A7.2 and A7.7.

Lemma A7.10 ($\R$ is an ordered field).

With Definitions A7.7 and A7.9, $\R$ satisfies the ordered-field axioms (Algebraic Structures), and the embedding $q \mapsto q^{*}$ is a field homomorphism. Rests on Definition A7.7, Definition A7.9, Lemma A7.8 and Definition 8.32.

Proof.

Derives Lemma A7.10. The verifications are of exactly the same kind as in Lemma A7.8 — elementwise inheritance from $\Q$ on the positive cone, then bookkeeping of the sign cases for associativity, commutativity, distributivity, and $\alpha\alpha^{-1} = 1^{*}$ (the reciprocal argument mirrors the additive-inverse argument with the multiplicative Archimedean walk $p, 2p, 4p, \dots$). They are long but routine, and no idea beyond those already displayed enters; we omit the repetition. Compatibility of order and product — $\alpha, \beta > 0^{*} \implies \alpha\beta > 0^{*}$ — is immediate from Equation (A7.1).

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Consequences

Corollary A7.11 (Archimedean property and density of $\Q$).

For $\alpha, \beta \in \R$ with $\alpha > 0^{*}$ there is $n \in \N$ with $n\alpha > \beta$; and between any two reals lies a rational. Rests on Theorem A7.6 and Definition A7.2.

Proof.

Derives Corollary A7.11. If $n\alpha \le \beta$ for all $n$, the set $A = \set{n\alpha \mid n \in \N}$ is bounded above, so $\gamma = \sup A$ exists (Theorem A7.6). Then $\gamma - \alpha < \gamma$ is not an upper bound: some $n\alpha > \gamma - \alpha$, whence $(n+1)\alpha > \gamma$ — contradicting that $\gamma$ bounds $A$. Density: if $\alpha < \beta$, pick $q \in \beta \setminus \alpha$; openness of $\beta$ gives $r \in \beta$, $r > q$, and then $\alpha \le q^{*} < r^{*} \le \beta$, so the embedded rationals $q^{*}, r^{*}$ separate $\alpha$ from $\beta$.

∎

Corollary A7.12 (Monotone convergence).

A nondecreasing sequence $(a_n)$ in $\R$ that is bounded above converges to $\sup_n a_n$. Rests on Theorem A7.6.

Proof.

Derives Corollary A7.12. Let $s = \sup_n a_n$, which exists by Theorem A7.6. Given $\varepsilon > 0$, $s - \varepsilon$ is not an upper bound, so $a_N > s - \varepsilon$ for some $N$; monotonicity gives $s - \varepsilon < a_n \le s$ for all $n \ge N$, i.e.\ $\abs{a_n - s} < \varepsilon$.

∎

Corollary A7.13 (Cauchy completeness).

Every Cauchy sequence $(a_n)$ in $\R$ converges in $\R$. Rests on Corollary A7.12 and Theorem A7.6.

Proof.

Derives Corollary A7.13. A Cauchy sequence is bounded (take $\varepsilon = 1$: beyond some $N$ all terms lie within $1$ of $a_N$, and only finitely many terms remain). Define $b_n = \inf_{k \ge n} a_k$, which exists because $\set{a_k \mid k \ge n}$ is bounded below and Theorem A7.6 applied to the negatives provides infima. The sequence $(b_n)$ is nondecreasing and bounded above, so by Corollary A7.12 it converges to $L = \sup_n b_n = \liminf_n a_n$. Given $\varepsilon > 0$ choose $N$ with $\abs{a_m - a_n} < \varepsilon/2$ for $m, n \ge N$. Then for $n \ge N$ every $a_k$ with $k \ge n$ lies in $(a_N - \varepsilon/2,\ a_N + \varepsilon/2)$, so $b_n \in [a_N - \varepsilon/2,\ a_N + \varepsilon/2]$ and hence $\abs{L - a_N} \le \varepsilon/2$; combining, $\abs{a_n - L} \le \abs{a_n - a_N} + \abs{a_N - L} < \varepsilon$ for $n \ge N$.

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Proof of Theorem A7.1. Derives Theorem A7.1. Assembling the section. The set of cuts of Definition A7.2, ordered by inclusion, is a totally ordered set containing an order-embedded copy of $\Q$ (Lemmas A7.3 and A7.5); it has the least-upper-bound property by Theorem A7.6; addition and multiplication of Definitions A7.7 and A7.9 make it an ordered field by Lemmas A7.8 and A7.10. The three remaining assertions are Corollary A7.11 (Archimedean, and $\Q$ dense) and Corollary A7.13 (Cauchy sequences converge). Existence is therefore established by construction, which is what the theorem claims.

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Remark A7.14 (Uniqueness).

Any two complete ordered fields are isomorphic as ordered fields: the isomorphism sends each element to the supremum of the embedded rationals below it, and Corollary A7.11 (which followed from completeness alone) makes the map well defined, bijective, and operation-preserving. Thus the real numbers are characterized by the axioms, not by the construction; Cantor's alternative construction by Cauchy sequences of rationals yields the same field. This is why Real Analysis may take Theorem A7.1 as its single axiom for $\R$ and never mention cuts again.