# proving an intersection empty set

by rbzima
Tags: intersection, proving
 P: 86 I'm trying to prove $$\bigcap^{\infty}_{n=1}(0,1/n) =$$ 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 $$n\in$$ N, assume we are given a closed interval $$I_n = [a_n, b_n] = {x \in$$ R $$: a_n \leq x \leq b_n}$$. Assume also that each $$I_n$$ contains $$I_n_+_1$$. Therefore, $$\bigcap^{\infty}_{n=1}I_n \neq$$ 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!
 P: 1,076 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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