Between two Rational Numbers exists Irrational Number/Lemma 2

Lemma for Between two Rational Numbers exists Irrational Number

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

Then:

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

Proof

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

By the definition of rational numbers:

$\exists p, q \in \Z: \alpha + \beta = \dfrac p q$

Thus:

$\beta = \dfrac p q - \alpha$

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

Therefore:

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

$\blacksquare$