Question on Real Numbers and Limit Point Prove that every uncountable subset of $R$ (real numbers), has a limit point.
I tried using Baire Category Theorem, which deals with uncountability, but I'm at sea.
If anyone can please help me with this problem I'll be glad. Thanks in advance

Write
$$\mathbb{R} = \bigcup\limits_{n=-\infty}^\infty [n,n+1).$$
Then think about whether every term in this union could have only finitely many points of the uncountable set in question, and what happens if one of them has more than finitely many.
