June 5th, 2018, 07:01 AM  #1 
Member Joined: Apr 2017 From: India Posts: 34 Thanks: 0  Closed set
The set S = {1(1/n): n belongs to N} is a closed set. True of False. Justify your answer. Just to be more informative. A set S is called a closed set if it contains all its limit points. Please help me with the answer. 
June 5th, 2018, 07:07 AM  #2 
Senior Member Joined: Aug 2012 Posts: 1,972 Thanks: 550 
Have you any thoughts on the matter? Why might it be a closed set? Why might it not be a closed set? If a set is closed if it contains all its limit points, what are the limit points of your set? Does your set contain them?

June 5th, 2018, 07:10 AM  #3 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 211 Thanks: 64 Math Focus: Algebraic Number Theory, Arithmetic Geometry 
Hint: what's the limit of $1  \frac{1}{n}$ as $n$ tends to infinity?

June 6th, 2018, 03:16 AM  #4 
Member Joined: Apr 2017 From: India Posts: 34 Thanks: 0 
It will be 1. But 1 does not belong to the set. All the very near points to 1 belong to S, but 1 itself does not belong to that. Is this the reason why it is not a closed set? Or is it a closed set as 1 belongs to it? Last edited by skipjack; June 6th, 2018 at 09:03 AM. 
June 17th, 2018, 11:12 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
You say "But 1 does not belong to the set" but then ask "Or is it a closed set as 1 belongs to it?"???


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closed, real analysis, set 
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