# 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.
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.
Suppose that $C$ is a subset of $\mathbb{R}$ with no limits point. Then for every $x\in\mathbb{R}$ you can find rational numbers $p_x$ and $q_x$ such that $p_x<x<q_x$ and $(p_x,q_x)\cap C$ contains at most one point. (Why?) How many intervals are there with rational endpoints? So how big can $C$ be?