Prove that every regular subset of $a^nb^n$ is finite
This is a simple application of the pumping lemma. Suppose that $L'$ is an infinite subset of $L$. Given $p$, since $L'$ is infinite, there exists some $n \geq p$ such that $w = a^nb^n \in L'$. Let $w = xyz$ be a decomposition of $w$ such that $|xy| \leq p$ and $y \neq \epsilon$. Then $y = a^t$ for some $t \neq 0$, and so $xy^0z = a^{n-t}b^n \notin L'$. Therefore $L'$ is not regular.
One might be tempted to think that this argument generalizes to every $L$ whose non-regularity can be proved using the pumping lemma. However, this is incorrect, as the example of $L \cup c^*$ shows.
This raises the following intriguing question:

For which infinite languages $L$ is it the case that all regular subsets of $L$ are finite?