Theorem
Letbe an open ball in the metric space
For every point
there exists an open ball
such that
hence
Define
We must show
Supposethen
or equivalently
hence
is a metric so
Hence
Theorem
Letbe an open ball in the metric space
For every point
there exists an open ball
such that
hence
Define
We must show
Supposethen
or equivalently
hence
is a metric so
Hence