I am trying to prove the following statement:
Set of all infinite subsets of natural numbers is equipotent with the power set of natutal numbers.
My thought is
Let the set of all infinite subsets of natural numbers be $S$.
We need to show that there is a bijection from $S$ to the power set of natural numbers. But I have no clue on how to start. Any help would be appreciated.
Best Answer
A different approach is to say that $S$ injects into $P(\Bbb N)$ because it is a subset. To inject $P(\Bbb N)$ into $S$, take any subset of $\Bbb N$ and double its members to get a set of even numbers. Take the union of that set with all the odd numbers. This is an infinite set of naturals, so is a member of $S$. We have injections both ways, so can use the Schroeder-Bernstein theorem to show there is a bijection.