Show that $L$ and $\overline L$ cannot be both finite 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 &quot;An Introduction to Formal Languages and Automata&quot; 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.
If $L$ and $\overline{L}$ were both finite, then so would $\Sigma^* = L \cup \overline{L}$ be, which we know is false.
This is a qualitative statement. We can make it quantitative by considering densities. Let us say that a language is dense if for infinitely many $n$, it contains at least half the words of length $n$. It is easy to check that at least one of $L,\overline{L}$ is dense.