I'm trying to show that any uncountable set has a countable subset. First, let me point out that the distinction here between at most countable and countable is applied in this instance. At most countable implies either finite or countable, and countable is obvious. Starting off, let X = the uncountable set, and choose some subset A of X, s.t. X\A = B. Not really sure where to go from this point. I had an idea that this could be proved inductively, but I'm not exactly sure how the process goes... Any suggestions?