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March 9th, 2016, 12:15 AM  #1 
Math Team Joined: Nov 2014 From: Australia Posts: 686 Thanks: 243  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. 
March 9th, 2016, 04:41 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,494 Thanks: 578 
You need "an infinite number of". I don't know how to reduce that to two words.

March 9th, 2016, 05:45 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,268 Thanks: 2434 Math Focus: Mainly analysis and algebra 
"infinitely many"?

March 9th, 2016, 06:37 PM  #4 
Math Team Joined: Nov 2014 From: Australia Posts: 686 Thanks: 243 
Can you justify it?

March 10th, 2016, 06:01 PM  #5 
Global Moderator Joined: May 2007 Posts: 6,494 Thanks: 578 
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 . 
March 10th, 2016, 08:01 PM  #6 
Math Team Joined: Nov 2014 From: Australia Posts: 686 Thanks: 243 
Yep. Spot on.


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infinite, intersections, unions 
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