The Real Numbers and Completeness

This is the first page of the analysis spine. Everything in calculus — limits, derivatives, integrals, and ultimately the integral theorems of vector calculus — rests on one structural fact that distinguishes $\mathbb{R}$ from $\mathbb{Q}$: completeness. This page states the field and order axioms, isolates the least-upper-bound axiom as the load-bearing assumption, and derives the consequences (Archimedean property, density, nested intervals) that the rest of the spine uses without comment.

References: Rudin, Principles of Mathematical Analysis, ch. 1; Tao, Analysis I.

1. Ordered Fields

A field is a set $F$ with two operations $+$ and $\cdot$ satisfying the usual axioms: $(F,+)$ is an abelian group with identity $0$; $(F\setminus \{0\},\cdot )$ is an abelian group with identity $1\ne 0$; and multiplication distributes over addition. $\mathbb{Q}$ and $\mathbb{R}$ are fields; $\mathbb{Z}$ is not (no multiplicative inverses).

An ordered field is a field $F$ with a total order $\le$ compatible with the operations:

• (O1) $x\le y⟹x+z\le y+z$ for all $z$;
• (O2) $x\ge 0$ and $y\ge 0⟹xy\ge 0$.

Both $\mathbb{Q}$ and $\mathbb{R}$ are ordered fields. The order lets us define the absolute value $|x|=\max (x,-x)$, which satisfies the triangle inequality $|x+y|\le |x|+|y|$ and $||x|-|y||\le |x-y|$.

Why $\mathbb{Q}$ is not enough. The equation $x^2=2$ has no solution in $\mathbb{Q}$, yet the set $\{q\in \mathbb{Q}:q^2<2\}$ is bounded above and "should" have a least upper bound. $\mathbb{Q}$ has holes. Completeness is the axiom that fills them.

2. Suprema, Infima, and the Completeness Axiom

Let $S\subseteq F$ be nonempty. An element $u\in F$ is an upper bound of $S$ if $x\le u$ for all $x\in S$; $S$ is bounded above if it has one. An upper bound $u$ is a least upper bound or supremum, written $\sup S$, if $u\le u^{\prime }$ for every upper bound $u^{\prime }$. Lower bounds and the infimum $\inf S$ are defined dually.

Characterization of the supremum. $u=\sup S$ iff (i) $x\le u$ for all $x\in S$, and (ii) for every $\epsilon >0$ there exists $x\in S$ with $x>u-\epsilon$. Condition (ii) says nothing smaller than $u$ is an upper bound.

Completeness Axiom (least-upper-bound property). Every nonempty subset of $\mathbb{R}$ that is bounded above has a supremum in $\mathbb{R}$.

This single axiom — false in $\mathbb{Q}$ — is what makes $\mathbb{R}$ the right setting for analysis. Taking $-S=\{-x:x\in S\}$ shows the dual statement (every nonempty set bounded below has an infimum) is equivalent.

$\mathbb{R}$ is the unique complete ordered field. Up to a unique order-preserving field isomorphism, $\mathbb{R}$ is the only ordered field satisfying the least-upper-bound property. This is why we may speak of "the" real numbers.

3. Constructing $\mathbb{R}$

The completeness axiom is consistent: one can build a complete ordered field from $\mathbb{Q}$. Two standard routes:

• Dedekind cuts. A real number is a partition $(A,B)$ of $\mathbb{Q}$ into a downward-closed set $A$ with no greatest element and its complement $B$. The cut for $\sqrt{2}$ is $A=\{q:q\le 0\text{ or }{q}^2<2\}$. The supremum of a bounded family of cuts is obtained by unioning their lower sets — completeness is almost immediate by construction.
• Cauchy completion. A real number is an equivalence class of Cauchy sequences of rationals, where $(a_n)\sim (b_n)$ iff $a_n-b_n\to 0$. This is the same metric-space completion construction used in general (see metric-spaces.md).

Both yield isomorphic structures, so the choice is a matter of taste. The analysis spine only ever uses the least-upper-bound property, never the internal details of the construction.

4. Consequences of Completeness

These corollaries are used constantly in later pages, usually silently.

4.1 The Archimedean property

Theorem (Archimedean). For every $x\in \mathbb{R}$ there is an $n\in >\mathbb{N}$ with $n>x$. Equivalently, for every $\epsilon >0$ there is $n\in \mathbb{N}$ with $1/n<\epsilon$.

Proof. If $\mathbb{N}$ were bounded above it would have a supremum $u$ by completeness. Then $u-1$ is not an upper bound, so $n>u-1$ for some $n$, giving $n+1>u$ — contradicting that $u$ bounds $\mathbb{N}$. $\square$

The Archimedean property fails in some ordered fields (e.g. fields with infinitesimals); completeness rules them out. It is the bridge that lets us control real quantities by integers, the engine behind every "choose $N$ large enough" argument in sequences-series.md.

4.2 Density of the rationals

Theorem. Between any two reals $x<y$ there is a rational $q$ with $x<q<y$ (and an irrational too).

Proof sketch. By Archimedes pick $n$ with $1/n<y-x$, then take $q=m/n$ for the least integer $m$ with $m/n>x$; one checks $m/n<y$. $\square$

Density is why continuous functions are determined by their values on $\mathbb{Q}$, and why decimal/rational approximation is legitimate.

4.3 Existence of roots

For every $a\ge 0$ and integer $n\ge 1$ there is a unique $x\ge 0$ with $x^n=a$, namely $x=\sup \{t\ge 0:t^n\le a\}$. The supremum exists by completeness; this is the prototype for "completeness produces the limit you expect to be there."

4.4 Nested interval theorem

Theorem (Nested intervals). If $I_1\supseteq I_2\supseteq \cdots$ are closed bounded intervals $I_n=[a_n,b_n]$, then ${\bigcap }_n{I}_n\ne >\varnothing$. If moreover $b_n-a_n\to 0$, the intersection is a single point.

Proof. The $a_n$ are increasing and bounded above by $b_1$, so $a=\sup a_n$ exists. For all $n$, $a_n\le a\le b_n$ (each $b_m$ is an upper bound of the $a_n$), hence $a\in \bigcap I_n$. If $b_n-a_n\to 0$ the point is unique. $\square$

This is the standard tool for "bisection" arguments and underlies the Bolzano–Weierstrass theorem and the proof of the Heine–Borel theorem in metric-spaces.md.

5. The Extended Reals

It is convenient to adjoin $\pm \infty$, forming $\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty ,+\infty \}$ with the conventions $-\infty <x<+\infty$ and $\sup \varnothing =-\infty$, $\sup S=+\infty$ when $S$ is unbounded above. With these conventions every subset has a supremum and infimum in $\overline{\mathbb{R}}$, which streamlines statements about $\mathrm{limsup}$/$\mathrm{liminf}$ in sequences-series.md. Arithmetic with $\infty$ is only partially defined ($\infty -\infty$ is indeterminate).

6. Where this page is used

• Completeness $\Rightarrow$ convergence: monotone convergence and Bolzano–Weierstrass in sequences-series.md.
• Archimedean property: every $\epsilon$–$N$ and $\epsilon$–$\delta$ argument in continuity.md and beyond.
• Suprema/infima: the definition of the Riemann integral via upper/lower (Darboux) sums in riemann-integral.md.
• Completeness of ${\mathbb{R}}^n$: the metric-space generalization in metric-spaces.md, which feeds the contraction-mapping proof of the inverse function theorem.

Next: Sequences and Series — convergence, Cauchy sequences, and the first payoffs of completeness.