I was trying some practice questions on Real Analysis. I can't seem to solve this one:
Let $ X := \{A \subset \Bbb{N} :$ both $A$ and $A^c$ are infinite$\}$. Prove that $X$ is uncountable.
From what I have studied about (un)countable sets is that the subset of a countable set can never be uncountable. It is known that $\Bbb{N}$ is a countable set. So how can such an $X$ exist which is a subset of $\Bbb{N}$ and is also uncountable?
Show the number of finite subsets of N is countable.
Show the number of subsets of N with finite complement is countable.
Show X is all the subsets of N not mentioned above.
Show there are uncountable many subsets of N.
Show us the rest of the proof that X is uncountable.
Riddle of the day: if a set is uncountable, does that mean it doesn't count?