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.

$\blacksquare$