Strongly decreasing sequence of sets
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?