I noticed that there is a question about $S$ being denumerable, which implies $S$ is equinumerous with a proper subset of itself, but what about an infinite set? That is, how to do I prove that every ...
I have the following decimal number: $0.01001100011100001111...$ Notice how whenever we have one 0, we also have one 1, two 0's, two 1's, etc. How do you continue it to infinity and prove that this ...
Proof: Between any two irrationals lies a rational, by the Density of the rationals in the real number system. There are only countably many rationals; therefore, there are only countably many pairs ...