1. The problem statement, all variables and given/known data Prove that any well-ordered subset (under the natural order) of the real numbers is countable. 2. Relevant equations None. 3. The attempt at a solution My attempt thus far has been to prove by contradiction. I didn't see a very clear way to get from well-ordered subset to countable so I decided to assume that I have a set A which is uncountable. The best I've been able to come up with is this: Since the set is uncountable and contained in R, it must contain at least one irrational number, and in fact, uncountably many irrationals. I recall another result, that is that the real line can only be covered by countably many disjoint intervals, meaning that at least one of these intervals contains uncountably many elements from my set A. Now, my intuition tells me that there is a way to essentially "shrink" this interval down to a very tiny neighborhood so that I can find my subset of A that is not well-ordered by the natural order as it behaves something like (0,1) or the like. I'm not sure how well this idea works, if at all, and was wondering if anyone thinks this idea is at all in the right direction or if there's another way to view the problem that might work better.