Theorem
Let
and
be open balls in a metric space
and let
then a ball open in
exists such that
Proof
Let
and
Suppose
so there exists
Since
there exists an open ball
and since
there exists an open ball
Take
then
and
so
Theorem
Letandbe open balls in a metric spaceand letthen a ball open inexists such that
Proof
LetandSupposeso there exists
Sincethere exists an open balland sincethere exists an open ball
Takethenandso