# Countably infinite set of real numbers with a complement that is infinite but not countably infinite

How can I show that if a set of real numbers is countably infinite, then its complement is infinite but not countably infinite?

Let the set you're looking for be $$\{a_1,a_2,a_3,\ldots\}.$$ Suppose the complement is countably infinite, so that it is $$\{b_1,b_2,b_3,\ldots\}.$$ Then consider the sequence $$\{c_1,c_2,c_3,c_4,c_5,c_6\ldots\}=\{a_1,b_1,a_2,b_2,a_3,b_3,\ldots\}.$$ That is countably infinite and contains all real numbers. But that's impossible. So the complement of a countably infinite set of reals relative to the set of all reals cannot be countably infinite.
HINT: If $|A|,|B|\leq\aleph_0$ then $|A\cup B|\leq\aleph_0$.