What happens if we divide $(0,1]$ into the union of countably many subsets?

Let $(0,1] \subset \mathbb{R}$ be the usual interval, where the left endpoint is open and the right endpoint is closed. Assume that$(0,1]$ is the union of countably many subsets $\left\{V_j\right\}_{j \in \mathbb{N}}$ with $V_j \subset(0,1]$.

Question: Can we find some $j_0 \in \mathbb{N}$ such that $V_{j_0} \cap(0,1 / k)$ is uncountable $\forall k \in \mathbb{N}$? If not, then can you give a counterexample?

Clues: If we replace $(0,1]$ with the compact set $[0,1]$, then the answer is yes (now the $V_j$ is subset of $[0,1]$). So, I guess the answer to this question is yes.