This was a homework question few weeks ago, and the teacher didn't give

out the solution to this problem (it doesn't directly concern complex analysis), so I was wondering weather someone knows the answer to this one.

Let

, and

be such that

is the set

with all isolated points of

removed. So the question is, is there a set

such that the sequence,

is strongly decreasing?

That is

,

, but not equal?

My first thoughts went something like this: Let

. Then

and

if

, so this wont do.

Now for every

we can make

-balls such that

if

and neither is 0. Then we can construct a sequence

for every

such that its limit is

and it is entirely contained in

. Lets call the set of those sequences

.

Now let

.

is such that

so now we got 4 strictly decreasing items in the sequence.

For every given

we can continue this construction

such that the sequence is strictly decreasing for the first

terms.

But that is not the same as the infinite sequence is strictly decreasing.

I am starting to think that this is impossible, but I am not sure.

Does anyone have any ideas?