Limit point proof

1. Oct 12, 2008

Unassuming

1. The problem statement, all variables and given/known data
I am trying to prove that every uncountable subset of R has a limit point in R.

(where R is the reals)

2. Relevant equations
I know that the reals are dense and you can find a real in between any two reals. I feel like if you keep finding a real in between two reals, you will find a limit point. I am not even sure where to start with this idea.

3. The attempt at a solution

2. Oct 12, 2008

e(ho0n3

3. Oct 12, 2008

Unassuming

Limit point of a set A: A limit point is a point x in which any neighborhood centered at x, no matter the size, intersects the set A at a point other than x.

The Bolzano-Weierstrass thrm states that , every bounded infinite subset of R has a limit point.

My problem does not have the bounded part, and is uncountable as opposed to the puny infinite.

I really don't understand why this works. Okay, if it's bounded and infinite then I feel convinced that it has a limit point (after seeing the thrm, of course). But non bounded, and uncountable? Is this problem stronger than the BW theorem?

4. Oct 12, 2008

jjou

Try using contradiction. Suppose your uncountable subset A in R does not have a limit point. That means, for any x in A, there is some ball around x which does not contain any element of A except itself. This should lead to a contradiction of A being uncountable.