Infinite prefix-closed context-free languages contain an infinite regular subset The Problem:

Say that a language is prefix-closed if all prefixes of every string
in the language are also in the language. Let C be an infinite,
prefix-closed, context-free language. Show that C contains an infinite
regular subset.

Can we show this by using Myhill-Nerode Theorem?
Given a normal form grammer $G$ for an infinite prefix-closed $L$, examine the (almost) regular grammer $G'$ obtained by transforming rules of the form $A\rightarrow BC$ into $A\rightarrow B$. I leave it to you to show that $L(G')$ satisfies your requirements.