Real Numbers are Densely Ordered
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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 densely ordered.
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 $(8)$