Infinite context free subset

Question

Disprove that:

Given an infinite recursive language $L$, there exists an context free, infinite subset $A$ of $L$.

This was asked in the exam. Any hint/partial solution?

Answer 1

Take the obviously recursive language $\{a^{n^2} \colon n \ge 1\}$. Using Parikh's theorem it is easy to see that no infinite subset can be context free (it can also be done with the pumping lemma, but that is more work).