Let $\{I\_k\}\_{k=1}^n$ be a finite collection of open intervals such that $\mathbb{Q} \cap [0,1] \subset \bigcup\_{k=1}^n I\_k$. How can I prove that the set $A=\{x\in \mathbb{I} \cap [0,1] : x \notin \bigcup\_{k=1}^n I\_k \}$ is finite?
($\mathbb{I}$ is the set of irrational numbers).