Sequence of a dense set

Question

Let $\mathbb{X}$ be a metric space with metric function $d_{\mathbb{X}}(\cdot,\cdot):\mathbb{X}\to\mathbb{R}_+$. Suppose $Y\subseteq\mathbb{X}$ is a dense set. Let $\{y_1,y_2,y_3,...\}\subsetneq Y$ be a countable sequence. Is it true that this sequence is also dense in $\mathbb{X}$? If not under what conditions would it be true?