Real Numbers are Densely Ordered/Proof 2
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Corollary to Between two Real Numbers exists Rational Number
- $\forall a, b \in \R: a < b \implies \paren {\exists c \in \R: a < c < b}$
Proof
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.
$\blacksquare$