- #1

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## Homework Statement

for every n in the natural numbers, Vn is a nonempty closed subset of a compact space T. Vn+1 is contained in Vn. Prove the Intersection of all the Vn's from n=1 to infinity does not equal the empty set.

## Homework Equations

## The Attempt at a Solution

This question seems rather easy, actually a little too easy for an advanced topology class. but this is what i did. I tried to prove by contradiction and assume that the Intersection of all the Vn's from n=1 to infinity EQUALS the empty set. By this assumption this would imply that there exists an n in the natural numbers such that that Vn is the empty set. I say this because... for any set A, A n {empty set} = {empty set}. so choosing a Vn to be the empty set would guarantee that the intersection of all the Vn's is the empty set. But this is a contradiction because in the original problem it says for every n in the natural numbers, Vn is NONEMPTY. Is this enough?