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

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?
