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Proving an intersection empty set 
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#1
Jan3008, 06:10 PM

P: 86

I'm trying to prove
[tex]\bigcap^{\infty}_{n=1}(0,1/n) = [/tex] EMPTY SET. One thing that I can't seem to bypass is getting past the closed intervals as stated in the Nested Interval Property, which states "For each [tex]n\in[/tex] N, assume we are given a closed interval [tex]I_n = [a_n, b_n] = {x \in[/tex] R [tex]: a_n \leq x \leq b_n}[/tex]. Assume also that each [tex]I_n[/tex] contains [tex]I_n_+_1[/tex]. Therefore, [tex]\bigcap^{\infty}_{n=1}I_n \neq [/tex] EMPTY SET. Basically, I'm stuck at taking care of the open interval part of this proof. The original for R was proved by showing sup A = x, then x is between the closed interval. Any advice would be great! 


#2
Jan3108, 12:16 AM

P: 1,075

Obviously the intersection is a subset of (0,1), Why? So it will suffice to show that for any x in (0,1) that there exists a set of the form (0, 1/n) to which x does not belong. So basically given x in (0,1) can you figure out how to find n such that x does not belong to (0, 1/n)?



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