how do you prove that U( n=1) ( -1/n, 1/n) ={0}?
first i tried to prove that RHS ⊂ LHS..
then to prove that LHS ⊂ RHS, i tried to do it in cases..
assume that x>0, then 1/x>0... so for 1/x ∈ ( -1/n, 1/n) , it means that 1/x < 1/n..then how do i prove by contraction if that is the case?
or is my method wrong?
No, there don't "have to be two sets". You can take the intersection of many sets!
This is the intersection of the open intervals (-1, 1), (-1/2, 1/2), (-1/3, 1/3), etc.
Alexandrabel90, RHS LHS should have been easy: 0 is in (-1/n, 1/n) for all n.
To prove LHS RHS, show that any x other than 0 is NOT in at least one of the intervals. If x is not 0 then it is either positive or negative. If x> 0, show that there exist some n such that 1/n< x (so x is not in (-1/n, 1/n) for that n). If x< 0, show that there exist some n such that x< -1/n (so x is not in (-1/n, 1/n) for that n.)
(Hint: Archimdean property)
Sigh, my mistake. Thank you HallsofIvy for helping me out.
"This is the intersection of the open intervals (-1, 1), (-1/2, 1/2), (-1/3, 1/3), etc."
This is the key point I wasn't understanding about the problem. You're right my wording was horrible about needing more than one set. Clearly subsets work. What I should have said was in what way intersections are being constructed isn't apparent to me.