Hello,
U is an open set in this topology. Let's assume that U is not the empty set.
Hence $\displaystyle \mathbb{R}\backslash U$ is finite and is a closed set of this topology.
Similarly, it is obvious that any closed set of this topology is finite, except $\displaystyle \mathbb{R}$, which is closed and open.
We have $\displaystyle U \cup \{\mathbb{R}\backslash U\}=\mathbb{R}$
But $\displaystyle \mathbb{R}$ is an infinite set, and $\displaystyle \mathbb{R}\backslash U$ is a finite set.
Thus U is an infinite set.
A definition of U being dense in $\displaystyle \mathbb{R}$, is that the unique closed set containing U is $\displaystyle \mathbb{R}$ itself.
But a closed set in the finite complement topology is, as said before, finite.
Thus there is no possible closed set containing U, except $\displaystyle \mathbb{R}$
This proves that U is dense in $\displaystyle \mathbb{R}$
Now you have to study the situation where U=$\displaystyle \emptyset$. It looks simple, but I'm quite confused... can you try it ?