.

. 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.