Let S be contained in $\displaystyle \mathbb{R} $ and let A be the set of accumulation points of S. Prove S is dense in $\displaystyle
\mathbb{R} $ iff. A=$\displaystyle
\mathbb{R}$
What is your definition of dense? Is it that for each ball in the reals, it contains a point of S?
If so then suppose that S is dense. Let $\displaystyle x \in \mathbb{R}$, then for each open set around x, it contains an element of S. Hence $\displaystyle x \in A$. As x was arbitrary, we have that $\displaystyle A=\mathbb{R}$.
Similarly the other way round.