Separable Subset of the Real Line Consider $\mathbb{R}$ with the usual topology. Let $A$ be an infinite subset of $\mathbb{R}$. Prove that $A$ is separable.
So, if $A$ is to be separable, it must contain a countably dense subset $H$. I need to show that this $H$ exists to prove that $A$ is separable. If I don't know more information about $A$, I am not sure how to do this.

Since $\mathbb{R}$ is second countable, it has a countable basis.  Intersecting each such basis element with $A$ is a basis of $A$ itself, so $A$ is also second countable.  The standard proof applies: for each basis element $B$, choose some $x_B\in B\cap A$, and let $D$ be the set of all such $x_B$.  This $D$ is countable, dense, and a subset of $A$.  To check that it is dense, for any open set $U\subset A$, there is some basis element $B$ inside it, and hence the point $x_B$, so $U\cap D$ is nonempty.
