Open intervals as union of closed intervals

Question

In Shreeve, it is stated that:

Since an open interval can be written as a sequence of closed intervals, $$(a,b)=\bigcup^\infty_{n=1}\left[a+\frac1n,b-\frac1n\right]$$

How can this be true?

Answer 1

$x \in (a,b) \iff a<x<b \iff$ there is $n \in \mathbb N$ such that $a+1/n \le x \le b-1/n \iff x \in [a+1/n,b-1/n] \iff x \in \bigcup_{n \in \mathbb N}[a+1/n,b-1/n].$