I'll first state the question as it appears:

Let $\displaystyle E \in \mathbb{R}^n$ and suppose that for every $\displaystyle x \in E, B(x;1) \cap E$ is countable. Prove that $\displaystyle E$ is countable.

To me: $\displaystyle E \in \mathbb{R}^n$, doesn't make sense, I'm interpreting this as one specific n-tuple, so I don't understand how an n-tuple can or cannot be countable. So I interpreted the question as $\displaystyle E \subset \mathbb{R}^n$. This may be wrong right away so if someone can explain to me why it should be the other way that would help.

But here's my attempt at a solution based on my interpretation:

I'm trying to use the Lindelof covering theorem which states that: Suppose $\displaystyle X \subseteq \mathbb{R}^n$. If $\displaystyle C$ is an open covering of $\displaystyle X$, then there is a countable subcollection $\displaystyle D \subseteq C$ that is also a cover of $\displaystyle X$. But I'm not able to make much progress with that method. And as of now I can't think of another way of showing it.

Any tips, or hints would be appreciated. Thanks in advance