Why can't the real set be an infinite reunion of closed sets?

Question

For homework, there was a question asking me to demonstrate that writing the Real set as: $\mathbb{R}$ = $\bigcup_n^\infty \mathbf{F_n}$, where $\mathbf{F_n}$ is a closed set, which does not contain an open set other than the empty set, is not possible. Frankly, I don't know where to start, as this construction of $\mathbb{R}$ seems perfectly fine.