How to prove that a set is not a finite union of intervals?

Question

Consider the set $\mathbb{R} - \mathbb{Z}$. It is certainly a countable union of intervals. How does one prove that it is not in fact a finite union of intervals?

Answer 1

A union of $n$ intervals would have at most $n$ connected components. $\mathbb R - \mathbb Z$ has infinitely many.

Answer 2

I’m assuming that by interval you mean bounded interval, i.e., sets of the forms $(a,b)$, $(a,b]$, $[a,b)$, and $[a,b]$. (Even if you allow rays, it’s not hard to show that $\Bbb R\setminus\Bbb Z$ doesn’t contain any rays.) Let $\{I_1,\ldots,I_n\}$ be any finite set of intervals, and for $n=1,\ldots,n$ let $a_n=\inf I_n$ and $b_n=\sup I_n$. Finally, let $b=\max\{b_1,\ldots,b_n\}$. Then $x\le b$ for each $x\in\bigcup_{k=1}^nI_k$, so $\lceil b\rceil+\frac12\in(\Bbb R\setminus\Bbb Z)\setminus\bigcup_{k=1}^nI_k$.