How can I prove that every uncountable subset of R has a limit point?

[Unassuming]

Homework Statement

I am trying to prove that every uncountable subset of R has a limit point in R.

(where R is the reals)

Homework 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.

The Attempt at a Solution

 

[e(ho0n3]

Start with the definition of limit point. What is it?

 

[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?

 

[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.