Show that if a,b Є R, and a≠b, then there exist ε-neighborhoods U of a and V of b of such that
(U intersection V)= Ø.
1
Expert's answer
2011-08-09T10:12:21-0400
For definiteness assume that a<b. Let e>0 be any number such that&
e < (b-a)/2
Denote U = (a-e,a+e), and V = (b-e,b+e),
Then U does not intersect V. To prove this it suffices to show that a+e < b-e which is equivalent to each of the following inequalities: 2e < b-a e < (b-a)/2
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