Let $L$ be any language on a nonempty alphabet. Show that $L$ and $\overline L$ cannot be both finite.
This is exercise 7 (page 28) from "An Introduction to Formal Languages and Automata" by Peter Linz.
My attempt: $L$ is a subset of $\Sigma^*$, so it can be either finite or infinite. If $L$ is finite, then $\overline L$ is infinite. If $L$ is infinite, that is, $L = \Sigma^*$, then $\overline L = \Sigma^* - L = \Sigma^* - \Sigma^* = \emptyset$. Therefore, $L$ and $\overline L$ cannot be both finite.
Is it correct? I tried to write a direct proof. Thanks in advance.