Existence of a convergent sequence implies that a set is dense

Question

Prove that, for all $A \subseteq\mathbb{R}$, $1$ implies $2$.

$1$. For every $x \in\mathbb{R}$, there is an infinite sequence $(a_n)$ with each $a_n \in A$, such that $\displaystyle\lim_{n\to\infty}a_n=x.$

$2$. $A$ is dense in $\mathbb{R}$.