$A$ is uncountable $\implies$ $A'$ is uncountable?

Question

For $A⊆\mathbb R$ , let $A'$ denote the set of all limit points of $A$ . If $A$ is uncountable , then does it necessarily mean that $A'$ is also uncountable ?

Answer 1

Yes, it does. In fact, $A\cap A'$ is necessarily uncountable; this answer contains a proof of that fact.