.
. So for
.
.
and since
is arbitrary,
, a contradiction.
(b) Since
is an isometry, it is clearly injective and continuous. If
is closed, then it is compact. Therefore,
is compact, which implies that
is closed (
is Hausdorff).
Suppose
is not surjective. Then let
. Since
is closed,
is open. Then for
, so for
,
.
Define a sequence as follows:
and let
I claim that
for
. Prove by induction.
Since
,
. Suppose it's true for the elements of
for
. We must show that
where
. It's true for
since
. If
, then
since
. Let
.
Then
is a discrete set. Since
is an infinite set with no limit point,
is not limit point compact, a contradiction.