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redyelloworange
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Homework Statement
Prove that every nonempty proper subset of Rn has a nonempty boundry.
The Attempt at a Solution
First of all, I let S be an nonempty subset of Rn and S does not equal Rn.
I tried to go about this in 2 different ways:
1) let x be in S and show that B(r,x) ∩ S ≠ ø and B(r,x) ∩ Sc≠ ø. I figured this wouldn't work with just one x in S. Or perhaps, I thought I should use induction on the number of elements in S?
2) Assume that bdS is empty and find a contradiction. However, I wasn't able to figure out a contradiction here. Unless, this implies that S equals Rn, then that's a contradiction. But I'm not quite sure it implies that. I think that this is the proof you use to show that Rn and the empty set are the only 2 that are both open and closed.
Thanks for your help! =)
Homework Statement
Homework Equations
The Attempt at a Solution
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