I need hand in the following question.
Where $A \subset \mathbb R$, show that the set of isolated points of A, is most countable.
My attempt was for this question, I followed the negation of definition of limit point. Then I claimed set of isolated points for $A$ is $E =\mathbb Q \cap A$. Then if $x$ is an isolated point of $A$, s.t. $x\in E$ then for all $r>0$, if $y$ is an element of $B_r(x)$, then we must have $x=y$.
To prove this I take for any $p/q$ element of $E$, $p$ and $q$ is integers such that with no common factors, then when we take $r= 1/q$, in ball $B_r(p/q)$, we only have $p/q$. I am wondering if this method is true. Or Am I doing something wrong here?