# Proving an intersection empty set

1. Jan 30, 2008

### rbzima

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!

2. Jan 31, 2008

### d_leet

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)?