Intersection of Sets

  • Thread starter levicivita
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  • #1

Homework Statement



let [itex]A_{1}, A_{2}, ....[/itex] be a sequence of subsets of R such that the intersection of the [itex]A_{1}, A_{2}, A_{3}...., A_{n} [/itex] is nonempty for each n greater than/equal to 1. Does it follow that the intersection of all [itex]A_{n}[/itex]'s is nonempty?
Does the answer change if you are given the extra information that each [itex]A_{n}[/itex] is a closed interval, that is a set of the form [itex][a_{n}, b_{n}] = {x member of R : a_{n} \leg x \leq b_{n}[/itex] for some pair of real numbers [itex](a_{n},b_{n})[/itex] with [itex]a_{n} < b_{n}[/itex]

Homework Equations





The Attempt at a Solution


I don't really have a clue how to start this. It seems to me that in both cases it should be non empty, but I'm really not sure. I'm not looking for some one to do this for me, because I want to be able to do it myself - I would appreciate it if someone could point me in the right direction without giving the game away.
 

Answers and Replies

  • #2
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For the first one, try to think of a counterexample. Perhaps try it for some sequence that converges to the empty set.

For the second part, you have a sequence of closed intervals such that the intersection of some finite collection of sets (in the sequence) is non empty. So what can you say about the intervals (think of them visually)?
 

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