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March 9th, 2016, 01:15 AM   #1
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Infinite intersections and infinite unions

I've started my study in a Bachelor of Science recently, and this question popped up on the second assignment for the maths course. I already know the answer, but I thought I'd post it for people to try.

Suppose we are given a set $A_n$ for every $n\in\mathbb{N}$.
Fill in the blank spaces with two words that make a true statement.

$\displaystyle x\in\bigcap_{k = 1}^\infty\bigcup_{n = k}^\infty A_n$ if and only if $x\in A_n$ for _______ _______ $n\in\mathbb{N}$.

The question didn't require a formal proof, but it did require some level of justification.
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March 9th, 2016, 05:41 PM   #2
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You need "an infinite number of". I don't know how to reduce that to two words.
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March 9th, 2016, 06:45 PM   #3
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"infinitely many"?
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March 9th, 2016, 07:37 PM   #4
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Can you justify it?
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March 10th, 2016, 07:01 PM   #5
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Let
a) If infinitely often, then for all k, and therefore .
b) If for only a finite number of n's, there will be a largest n, say m so that for k > m, and therefore .
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March 10th, 2016, 09:01 PM   #6
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Yep. Spot on.
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