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Homework Statement
suppose that a metric space A is a union A = B U C of two subsets of finite diameter. Prove A has finite diameter.
Homework Equations
The Diameter of a metric space M is sup D(a,b) for all a,b in M.
The Attempt at a Solution
Really, no idea where to begin. I just began this subject, and I'm not too well versed in thinking of what these objects really are. If I'm understanding this right, I can imagine having say two disjoint rectangles that are far apart, and these are my B and C, with A being the union of them. Then each B and C may have a diameter, which represents the maximum distance between any two points in them, and part of me wants to take the farthest point of one and connect it to the other, but that seems like that wouldn't always work.
The only thing I can think of is that since both sets have finite diameter, then neither set is infinite. Therefore there must exist a set of points that are the farthest apart...
But like I said, I'm new to this subject (metric spaces) and have no idea where to begin. Any help would be greatly appreciated. (In case it matters, I haven't studied topology, and balls begin in the next section (although I have learned a little about neighborhoods, not sure if that's the same thing)