1. The problem statement, all variables and given/known data Call a set X Dedekind infinite if there is a 1-to-1 mapping of X onto its proper subset. Prove that every countable set is Dedekind infinite. 3. The attempt at a solution I want to say that every countable set can be well ordered. I guess I could just pick some element from our set X and call it a. And then make sure everything from our set gets mapped to something larger than a. So we have a 1-to-1 mapping to our proper subset. I probably need to be more rigorous about how this mapping takes place.