Taking away infinitely many elements infinitely many times This is a somewhat hand wavy question but I'm not sure how to ask it more precisely.
If we have a countably infinite sequence (or set), can we take away infinitely many elements from the sequence infinitely many times and still be left with infinitely many elements? Or is it the case that we must be left with 0 or finitely many elements?

Yes, this is possible.
For example, consider the set of all natural numbers $\mathbb N$. In your first move, remove every number that is a power of $2$. In your second move, remove (from the remaining set) every number that is a power of $3$. In your $n$-th move remove all numbers that are a power of the $n$-th prime.
After you have done this infinitely many times (once for each prime), there are still infinitely many numbers left - namely those with at least 2 distinct prime factors (and also the number 1).
