I've been going over previous exams that I've had in topology to study for my comprehensive exam, and I noticed a problem that I missed. I was wondering if anyone could help me out with this problem:

Let $X$ be a Lindelöf topological space and let $A \subset X$. Suppose that every $x \in X$ has a neighborhood $V$ such that $V \cap A$ is countable. Prove that $A$ is countable.

Thanks in advance for any help.