Between two Real Numbers exists Rational Number/Proof 2

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Theorem

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

Then:

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


Proof

As $a < b$ it follows that $a \ne b$ and so $b - a \ne 0$.

Thus:

$\dfrac 1 {b - a} \in \R$

By the Archimedean Principle:

$\exists n \in \N: n > \dfrac 1 {b - a}$

Let $M := \set {x \in \Z: x > a n}$.

By Set of Integers Bounded Below has Smallest Element, there exists $m \in \Z$ such that $m$ is the smallest element of $M$.

That is:

$m > a n$

and, by definition of smallest element:

$m - 1 \le a n$


As $n > \dfrac 1 {b - a}$, it follows from Ordering of Reciprocals that:

$\dfrac 1 n < b - a$

Thus:

\(\ds m - 1\) \(\le\) \(\ds a n\)
\(\ds \leadsto \ \ \) \(\ds m\) \(\le\) \(\ds a n + 1\)
\(\ds \leadsto \ \ \) \(\ds \frac m n\) \(\le\) \(\ds a + \frac 1 n\)
\(\ds \) \(<\) \(\ds a + \paren {b - a}\)
\(\ds \) \(=\) \(\ds b\)


Thus we have shown that $a < \dfrac m n < b$.

That is:

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

such that $r = \dfrac m n$.

$\blacksquare$


Sources

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