[SOLVED] Real Analysis Proof: Intersection and Union of an Indexed Set

For each http://qaboard.cramster.com/Answer-B...2087503987.gif let http://qaboard.cramster.com/Answer-B...0833750886.gif. (Note Q is the set of all rationals).

Prove that:

(1) http://qaboard.cramster.com/Answer-B...2712509613.gif (that is the intersection over the entire index is the singleton set with 0).

and prove that

(2) http://qaboard.cramster.com/Answer-B...8025003118.gif (that is the union over the entire index is the set of all rationals in the interval [0,1) ).

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Okay, wow. I am utterly stumped by this. Things I do know that $\displaystyle E_x$ is the set of all sets with

the form [0 , x) for all x in the interval (0, 1). The intersection is what all of those sets have in common, and I can see that it is 0; but I can't figure out the proof. Since it is proving two sets equal, I would assume having to "chase elements" and prove the set on the right side is a subset of the left and vice versa. We did a proof similar to this in class, and our professor proved it using contradiction. So then maybe I will also need to use contradiction to prove these two statements?

As for the union proof, I'm also thinking of a chasing elements proof. But again, where to start is what I am struggling with.

I listed out some sets like:

$\displaystyle E_{1/16}$ = [0, 1/16)

$\displaystyle E_{1/8}$ = [0, 1/8)

$\displaystyle E_{1/4}$ = [0, 1/4)

$\displaystyle E_{1/3}$ = [0, 1/3)

$\displaystyle E_{1/2}$ = [0, 1/2)

So now I kind of 'see' better how the union would be everything including 0 but not reaching 1 because x $\displaystyle \in$ (0, 1).

Any helps, hints, tips, and/or suggestions on getting started are greatly appreciated!

Thank you for your time.