Prove that an uncountable set of $\displaystyle \bold{R} $ must have a limit point.

If a set $\displaystyle S \subset \bold{R} $ and is uncountable, then $\displaystyle \aleph_{0} < \text{card}(S) $. Now a point $\displaystyle x_0 $ is a limit point of $\displaystyle S \subset \bold{R} $ if for every $\displaystyle \varepsilon > 0 $, $\displaystyle (x_{0}- \varepsilon, \ x_{0}+ \varepsilon) \cap S \ \backslash \{x_0 \} \neq \emptyset $.

Suppose for contradiction that $\displaystyle S $ didnothave a limit point. Would this then create a gap, contradicting the fact that the set is uncountable?