Question #3880

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.

Expert's answer

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.

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.

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