# Homework Help: Prove intersection is empty

1. Feb 2, 2012

### cragar

1. The problem statement, all variables and given/known data
Prove that $\bigcap_{n=0}^{\inf} (0,\frac{1}{n})=\emptyset$
3. The attempt at a solution
since 0 is not included in our interval. eventually I will get to
(0,0) because I could pick a real as close to zero as I wanted and there would be a natural such that $\frac{1}{n}<y$ therefore this intersection is empty.
but if my orginal intersection was [0,1/n] then this would not be empty.

2. Feb 2, 2012

### jbunniii

You have the right idea, and you're correct that if (0,1/n) was replaced by [0,1/n], the intersection would not be empty. It would contain exactly one point: 0.

You could word your proof a bit precisely as follows:

Suppose the intersection is non-empty. Then there exists some x in the intersection:

$$x \in \bigcap_{n=1}^{\infty} (0, 1/n)$$

Since x is in the intersection, it means that x must be in every interval of the form (0, 1/n), and this is impossible because...

3. Feb 2, 2012

### cragar

ok thanks for the input

4. Feb 3, 2012

### Deveno

you might want to show that if x > 0, there exists k in N with 1/k < x, first.

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