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.Q:
Proof that every subset of $\mathbb{R}$ is countably infinite.
I have to prove that every subset of $\mathbb{R}$ is countably infinite.
I started with assuming that there is a subset $U$ of $\mathbb{R}$ which is not countably infinite, but I don’t know how to proceed.
Any help is appreciated.
Thanks in advance.
A:
Let the set $S$ be as finite as possible and let $s\in S$. If $S$ is non-empty, $S$ must have a minimal element (minumum with respect to set inclusion). If $S$ is countable, $S$ cannot have a minimal element, which we have assumed to be a contradiction.
A:
One way to do it is to take $U$ to be any non-empty set. Let $\{U_i:i\in I\}$ be a partition of $U$ into infinite sets, and observe that the union of these sets is a countable union of countable sets, and hence countable.
A:
Can you guess why not?
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