Real Numbers are Densely Ordered/Proof 1
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Theorem
- $\forall a, b \in \R: a < b \implies \paren {\exists c \in \R: a < c < b}$
Proof
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 $b$.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: Exercise $\S 1.8 \ (6)$