Let $A\subseteq\mathbb{R}$ with $A$ uncountable.
Let $B\subseteq A$ where for every $b\in B$, $\exists n\in\mathbb{N}$ where $\left[b,b+\frac1n\right)\cap A$ is countable.
Now, show $B$ is countable.
I was thinking of assuming $B$ is uncountable and taking one interval for every $b\in B$. Then I would take a rational from each interval, and derive a contradiction as there are only countably many rationals. This doesn't work as the intervals could overlap. I'm unsure how to continue.