Between two Rational Numbers exists Irrational Number/Lemma 1

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Lemma for Between two Rational Numbers exists Irrational Number

Let $\alpha \in \Q \setminus \set 0$ and $\beta \in \R \setminus \Q$.

Then:

$\alpha \cdot \beta \in \R \setminus \Q$


Proof

Aiming for a contradiction, suppose $\alpha \cdot \beta \in \Q$.

By the definition of rational numbers:

$\exists n, m, p, q \in \Z: \alpha = \dfrac n m$
$\alpha \cdot \beta = \dfrac p q$

Thus:

$\beta = \dfrac p q \cdot \dfrac 1 \alpha = \dfrac p q \cdot \dfrac m n$


By Rational Multiplication is Closed, we have $\beta \in \Q$, which contradicts the statement that $\beta \in \R \setminus \Q$.


Therefore $\alpha \cdot \beta \in \R \setminus \Q$.

$\blacksquare$