Reals are Close Packed

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Theorem

Let $a, b \in \R$ such that $a < b$.

Then:

$\exists c \in \R: a < c < b$


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 Every Two Reals Exists a Rational:

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

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

$\blacksquare$