R/Q is Dense in R

Theorem

$\mathbb{R}-\mathbb{Q}$ is dense in $\mathbb{R}$. If $a \in \mathbb{R}$, $b \in \mathbb{R}$, and $a < b$, then there exists an irrational number $r \in \mathbb{R}-\mathbb{Q}$ such that $$ a < r < b. $$

Strategy

We want to use the density of $\mathbb{Q}$ theorem that we proved here.

Formal Proof

[TODOOOOO]

References

Lecture Notes by Professor Chun Kit Lai