Show that every infinite language has a non-regular subset I'm trying to solve this problem:


  Let $L$ be some infinite language, show that there exists a sub-language of $L$ that is not regular


But can this be correct? If I have the language $\{a\}^*$ for example, that's infinite but you can make a DFA for any sub-language of it, right?

There's a hint that this can be proved using diagonalization, but I think I must be misunderstanding the question.
The powerset of the language contains all the sublanguages. This will be uncountably infinite, but there are only countably many regular languages (for a particular, finite alphabet)