Sunday, September 22, 2019

Neighborhoods in real analysis problem



Show that if a,bϵR, then there exists ε neighborhoods U of a and V of b such that UV=.



I have already defined the sets Vε(a):={xϵR:|xa|<ε} and Uε(b):={yϵR:|yb|<ε} but I don't know how to proceed further. Any help would be appreciated.


Answer



Draw a picture. If ϵ=|ba|3, it is clear that the intervals (aϵ,a+ϵ) and (bϵ,b+ϵ) have no point in common.




If we want to be very formal, suppose to the contrary that |bx|<ϵ and |xa|<ϵ. Then by the Triangle Inequality
|ba||bx|+|xa|<2ϵ<|ba|,


which is impossible.


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