My Math Forum Infinite intersections and infinite unions

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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. Thanks from greg1313 and Joppy
 March 9th, 2016, 04:41 PM #2 Global Moderator   Joined: May 2007 Posts: 6,528 Thanks: 589 You need "an infinite number of". I don't know how to reduce that to two words. Thanks from greg1313
 March 9th, 2016, 05:45 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,313 Thanks: 2447 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: 6,528 Thanks: 589 Let $S_k=\cup_{n=k}^{\infty}{A_n}.$ a) If $x\in{A_n}$ infinitely often, then $x\in{S_k}$ for all k, and therefore $x\in {\cap_{k=1}^{\infty}{S_k}}$. b) If $x\in{A_n}$ for only a finite number of n's, there will be a largest n, say m so that $x\notin{S_k}$ for k > m, and therefore $x\notin{\cap_{k=1}^{\infty}{S_k}}$.
 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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