TITLE: Taking away infinitely many elements infinitely many times
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: Other possibilities: 

Take the natural numbers. First remove the numbers that are divisible by 2, but not by 4. Then remove the numbers that are divisible by 4, but not by 8. Then remove the numbers that are divisible by 8, but not by 16. In general, in the $n$th step, remove the numbers that are divisible by $2^n$, but not by $2^{n+1}$. You are left with the odd numbers. 
Take the rational numbers. At the $n$th step, remove all rationals that are between $n-1$ and $n$. You are left with the negative rational numbers.