Question

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.

Answer 1

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

But the last
inequality holds by assumption.