= Completeness of the real numbers
{wiki}
Every <Cauchy sequence> of <real numbers> converges to a <real number>. Explicitly, the condition
$$
\forall\varepsilon>0\;\exists N\in\mathbb N_{>0}\;\forall m,n\geq N:\quad |x_m-x_n|<\varepsilon
$$
implies the existence of $L\in\mathbb R$ with $x_n\to L$. The hypothesis compares approximations with one another, without requiring a candidate answer. This makes <completeness of the real numbers> an existence principle: sufficiently consistent approximations cannot approach a missing destination. https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_lec102.pdf[MIT 18.100A, Lecture 10, theorem 7]{ref}.
The equivalent <least-upper-bound property> says that every nonempty <set> $A\subseteq\mathbb R$ bounded above has a <supremum> $s\in\mathbb R$. Precisely, every $x\in A$ satisfies $x\leq s$, and for every $\varepsilon>0$ some $x\in A$ satisfies $s-\varepsilon<x$. The second condition excludes any smaller <upper bound>. No largest element of $A$ is required.
The <rational numbers> lack this property. The <set> $A=\{q\in\mathbb Q:q\geq0,\ q^2<2\}$ contains one and has <upper bound> two, but has no <supremum> in $\mathbb Q$. A proposed rational <supremum> $s$ would have to satisfy $s^2=2$: if $s^2<2$, a sufficiently small rational increase still lies in $A$; if $s^2>2$, a sufficiently small rational decrease remains an <upper bound>. The <irrationality of the square root of two> excludes equality. In $\mathbb R$, its <supremum> is $\sqrt2$. https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_lec32.pdf[MIT 18.100A, Lecture 3, theorems 14–15]{ref}.
Construct this <square root> by the <bisection method>, starting with rational bounds $[l_0,u_0]=[1,2]$. At each step square the midpoint and retain the half whose endpoints bracket two after squaring. The first four resulting <intervals> are $[1,3/2]$, $[5/4,3/2]$, $[11/8,3/2]$, and $[11/8,23/16]$. After $k$ steps, $u_k-l_k=2^{-k}$ and $l_k^2<2<u_k^2$. Both endpoint <sequences> are <Cauchy sequences> because every later endpoint remains in the current <interval>. <Completeness of the real numbers> gives a common <limit of a sequence> $L$; <continuity> of squaring gives $L^2=2$. The midpoint approximates $L$ with error at most $2^{-(k+1)}$. Thus exact rational arithmetic produces certified accuracy while <completeness of the real numbers> supplies the exact answer.
<Richard Dedekind> approached <completeness of the real numbers> by filling gaps in the <rational numbers> with <Dedekind cuts>. In the preface to his 1872 Continuity and Irrational Numbers, he traced the problem to teaching <differential calculus> in 1858 and wanting arithmetic foundations for <limits>. His construction makes a missing boundary, such as $\sqrt2$, into a <number> defined by the separation it produces among <rational numbers>. https://www.gutenberg.org/files/21016/21016-pdf.pdf[Dedekind, Continuity and Irrational Numbers, preface and §§III–IV; Beman's 1901 translation]{ref}.
Back to article page
