# Real Numbers are Close Packed

Jump to navigation
Jump to search

## Contents

## Corollary to Between two Real Numbers exists Rational Number

- $\forall a, b \in \R: a < b \implies \paren {\exists c \in \R: a < c < b}$

That is, the set of real numbers is close packed.

## Proof 1

We can express $a$ and $b$ as:

- $a = \dfrac a 1, b = \dfrac b 1$

Thus from Mediant is Between:

- $\dfrac a 1 < \dfrac {a + b} {1 + 1} < \dfrac b 1$

Hence $c = \dfrac {a + b} 2$ is an element of $\R$ between $a$ and $b$.

$\blacksquare$

## Proof 2

From Between two Real Numbers exists Rational Number:

- $\exists r \in \Q: a < r < b$

Since a rational number is also a real number, the result follows by definition.

$\blacksquare$

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(8)$