A question about countability of a set Let $\left\{ x_\alpha : \alpha \in \mathscr{A}\right\} \subset (0, + \infty ) $ be a set of positive real numbers such that for every countable subcollection $ \left\{ x_{\alpha_n} \right\} $ of distinct points it holds $ x_{\alpha_n} \rightarrow 0 $. Then $ \mathscr{A} $ is a countable set. \
I think that this statement is true. How can i prove it? (if it is true)
Thanks

Hint: for each $n$, there are only finitely many $x \in \mathcal{A}$ such that $|x| \geq \frac{1}{n}$ -- otherwise you can easily find a sequence in $\mathcal{A}$ that does not converge to 0.
