October 19th, 2016, 11:12 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 861 Thanks: 68  Nested open intervals
Why does the nested interval theorem require closed intervals? Let ai and bi converge to a. Lim (ai,bi)=(a,a) doesn't exist: a<x<a Lim (ai,bi]=(a,a] doesn't exist: a$\displaystyle \leq$x<a Lim [ai,bi]=[a,a] exists: a$\displaystyle \leq$x$\displaystyle \leq$a, x=a 
October 19th, 2016, 01:33 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,173 Thanks: 477 
Since ai and bi both converge to a, then a is in the limit interval. (a,a) and (a,a] are both empty (do not contain a), so they are wrong.


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