Show that if a,bϵR, then there exists ε neighborhoods U of a and V of b such that U∩V=∅.
I have already defined the sets Vε(a):={xϵR:|x−a|<ε} and Uε(b):={yϵR:|y−b|<ε} but I don't know how to proceed further. Any help would be appreciated.
Answer
Draw a picture. If ϵ=|b−a|3, it is clear that the intervals (a−ϵ,a+ϵ) and (b−ϵ,b+ϵ) have no point in common.
If we want to be very formal, suppose to the contrary that |b−x|<ϵ and |x−a|<ϵ. Then by the Triangle Inequality
|b−a|≤|b−x|+|x−a|<2ϵ<|b−a|,
which is impossible.
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