Example of a non-regular language that is a subset of a regular language?

Question

How can this be? I don't think it is actually possible for a non-regular language to be a subset of a regular language. What examples are there where this is true?

Answer 1

Every language over an alphabet $\Sigma$ is, by definition, a subset of $\Sigma^*$, which is regular. If you want a less trivial example,

$$\{a^nb^n\mid n\geq 0\}\subseteq L(a^*b^*)\,.$$