X uncountable implies X' uncountable

Question

Let $X \subseteq \mathbb{R}$ an uncountable set. So the set of limits points ($X'$) is uncountable.

I understand this intuitively but I don't know how the information "$X$ is uncountable" is useful to get informations about $X'$

Answer 1

For every point $a\in X\setminus X'$, thee is some $r>0$ such that $B_r(a)\cap X= \{a\}$. The sets $B_{r/2}(a)$ are pairwise disjoint. Pick a rational from each $B_{r/2}(a)$. In the end, you pick $|X\setminus X'|$-many rationals.