1. The problem statement, all variables and given/known data If each of D1 and D2 is a subdivision of [a,b], then... 1. D1 u D2 is a subdivision of [a,b], and 2. D1 u D2 is a refinement of D1. 2. Relevant equations **Definition 1: The statement that D is a subdivision of the interval [a,b] means... 1. D is a finite subset of [a,b], and 2. each of a and b belongs to D. **Definition 2: The statement that K is a refinement of the subdivision D means... 1. K is a subdivision of [a,b], and 2. D is a subset of K. 3. The attempt at a solution I just proved that, "If K is a refinement of H and H is a refinement of the subdivision D of [a,b], then K is a refinement of D." Well I havent wrote it down but the 2nd definition part 2 is what makes it easy to relate just being a transitive proof. My problem is that I've taken a lot of logic courses in the past so when I see the union of two variables I only need to prove that one is actually true. In this particular situation both are true so its obvious but I don't know how to state that fact. For the 2nd part of the proof, wouldn't I just say that D1 is a subset of itself, and its already given that D1 is a subdivision of [a,b]? It just seems too easy... I also had questions about proofs I've already turned in that I did poorly on but I didn't want to flood this place with questions.