# Proving the intersection and union of the indexed...

• May 5th 2009, 11:52 AM
Danneedshelp
Proving the intersection and union of the indexed...
For every n∈N, let An=[0,1/n], closed interval of the real line. Find the union and the intersection of the indexed family {An}n∈N. Justify your answer.

Well, I got the union:

∪F=[0,1] (where F=the above indexed family)

and the intersection:

∩F={0}

I am having troubles proving this.

For the union I choose an arbitrary x∈∪F. x is an element of ∪F ⇔ ∃n∈N(x∈An). Clearly, if x∈An for some n∈N ⇒ x∈[0,1] for some n∈N. So, 0<x<1/n≤1.
Now, for the second inclusion we choose some x∈[0,1]....?

I don't really know how to prove either. My teacher says the archimedean principle should be in the proof. I have my final tomorrow and wouldl appreciate some help with this. It's one of the concepts I never really got in the class.

Thanks.
• May 5th 2009, 01:20 PM
Plato
There is nothing to prove about union.
$A_1=[0,1]$ so the union is $[0,1]$.

If $0 then $\left( {\exists n} \right)\left[ {\frac{1}
{n} < x} \right]\; \Rightarrow \;x \notin A_n \; \Rightarrow \;x\notin\bigcap\limits_n {A_n }$

But $\left( {\forall n} \right)\left[ {0 \in A_n } \right]\; \Rightarrow \;0 \in \bigcap\limits_n {A_n }$