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October 19th, 2016, 11:12 AM   #1
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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
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October 19th, 2016, 01:33 PM   #2
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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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