Closure of the Reciprocal Sequence in a Metric Space
Theorem
Let
be the space of real numbers in the metric space
with the absolute value metric
and let
The closure of
is
and the closure of any subsetof
is a closed subset of
Proof
Let
The closure of
written
is defined as
where 
Since
For each
Hence
and
Choose
and
then
hence
To prove the closure of any subsetofis a closed subset of
supposeis not closed then
is not open and there is an element
such that for every
Choose
Since
is open there exists
such that
Since
then
so
exists such that
Hence for
there existssuch that
Hence contradicting
