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Math 2.0

Real functions Proof

Proof of the density of the irrationals

Builds on Density of the rationals Existence of square roots of nonnegative reals Irrationality of the square root of 2 Irrational number Rational numbers The reals form an ordered field

Proof

Let $x$ and $y$ be real numbers with $x < y$ .
By the existence of square roots, there is a real $s \ge 0$ with $s^2 = 2$ . Since $s^2 \ne 0$ , we have $s \ne 0$ , so $s > 0$ .
The number $s$ is irrational, by the irrationality of the square root of 2.
Dividing by $s > 0$ keeps the order in the ordered field $\R$ . So $x/s < y/s$ .
By the density of the rationals, there is a rational $q_1$ with $x/s < q_1 < y/s$ . Again by density, there is a rational $q_2$ with $q_1 < q_2 < y/s$ .
Since $q_1 \ne q_2$ , at least one of them is nonzero. Call it $q$ . Then $q$ is rational, $q \ne 0$ and $x/s < q < y/s$ .
Multiply by $s > 0$ : $x < q s < y$ . Let $z = q s$ .
Suppose $z$ is a rational number $r$ . Then $s = r/q$ is a quotient of rationals with $q \ne 0$ , so $s$ is rational. This contradicts step 3.
So $z$ is an irrational number with $x < z < y$ .

Idea

A nonzero rational multiple of $\sqrt{2}$ is irrational. Scale the interval by $1/\sqrt{2}$ , find a nonzero rational inside it, and scale back.

Reading path

To understand this page from the ground up, read these 79 pages in order. The foundations come first. The path takes the proofs Proof of Bézout's identity, Proof of the approximation property of the supremum, Proof of the generalized Euclid's lemma, Proof of the commutativity of addition by induction, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the irrationality of the square root of 2 by lowest terms, Proof of Euclid's lemma, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of strong induction from ordinary induction, Proof of the density of the rationals, Proof of the existence of square roots of nonnegative reals, Proof of the existence and uniqueness of a fraction in lowest terms, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof that the reals are Archimedean from the completeness axiom, Proof of the characterization of the floor.

Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Divisibility definition
Element definition
Integers object
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Set bounded above definition
Greatest common divisor definition
Prime number definition
Set definition
Ring of integers statement
Archimedean ordered field definition
Successor function definition
Implication definition
Predicate definition
Irrational number definition
The reals form a field statement
Supremum definition
Coprime integers definition
Ordered pair definition
Addition of natural numbers definition
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Universal quantifier definition
Completeness axiom statement
Cartesian product definition
Order on the natural numbers definition
Proof of the commutativity of addition by induction proof
Subset definition
The rationals form an ordered field statement
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Strong induction statement
Preorder definition
Proof of the well-ordering principle from strong induction proof
Partial order definition
Well-ordering principle statement
Total order definition
Proof of the division algorithm proof
Division algorithm statement
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Proof of Bézout's identity proof
The reals form an ordered field statement
Bézout's identity statement
Proof of the existence of square roots of nonnegative reals proof
Proof of the approximation property of the supremum proof
Proof of the generalized Euclid's lemma proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Generalized Euclid's lemma statement
Square root definition
Proof that the reals are Archimedean from the completeness axiom proof
Proof of Euclid's lemma proof
Proof of the existence and uniqueness of a fraction in lowest terms proof
The reals are Archimedean statement
Euclid's lemma statement
Floor function definition
Proof that there is a natural number n with 1/n below any positive epsilon proof
Fraction in lowest terms statement
Proof of the characterization of the floor proof
There is a natural number n with 1/n below any positive epsilon statement
Proof of the irrationality of the square root of 2 by lowest terms proof
Characterization of the floor statement
Irrationality of the square root of 2 statement
Proof of the density of the rationals proof
Density of the rationals statement
Proof of the density of the irrationals proof

Where this leads

1 page builds on this one.

Density of the irrationals

Further reading

Web https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Irrational_Number

About this page

Tag	`JR1Z7GC7J0`
URI	`m2://JR1Z7GC7J0`
Kind	proof
Writing	W0 stub
MSC	26A03
Proves	Density of the irrationals
Uses	Density of the rationals, Existence of square roots of nonnegative reals, Irrationality of the square root of 2, Irrational number, Rational numbers, The reals form an ordered field
Changed	4 November 2026

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