Limit point and dense sets in $\mathbb R$ If every real number is a limit point of $A ⊆$ $\mathbb R$ then is it true that $A$ is dense in  $\mathbb R$ ? 

Yes, indeed. A set $A\subseteq\Bbb R$ is dense in $\Bbb R$ if and only if its closure is all of $\Bbb R.$ If every point of $\Bbb R$ is a limit point of $A$, then every point of $\Bbb R$ is in its closure, so $A$ is dense in $\Bbb R.$
