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Mathematics
Calculus
Proving half of the Heine-Borel theorem
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[QUOTE="Office_Shredder, post: 6573723, member: 53426"] A couple notes. 1.) You don't need to know that X is closed to know the set of limit points is bounded. Your argument is right, but just knowing X is bounded is sufficient as well (if it's not clear why, this is a good exercise). 2.) Your hammering away at the collection of U"s doesn't work. I think your mental model should be that things that look like paragraph 2 are always wrong. Consider ##X=[0,1]## vs ##X=[0,1]\cup[2,3]## I could hand you an open cover, and you could go through your whole procedure on the latter one, and only end up with a cover of [0,1]. You would then need to do it again to get a cover of the [2,3] part. I also think you don't need it. Paragraph 3 is the right place to start. If you have an open cover with no finite sub over, then you can construct a countable sequence of points from it, even if the original cover was uncountably large. That sequence doesn't necessarily converge to the supremum, but what can you say about convergence? [/QUOTE]
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Forums
Mathematics
Calculus
Proving half of the Heine-Borel theorem
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