Problem:

For a language L, Substring(L) = {w : xwy L}. In other words, Substring(L) consists of any word that can be created by taking a word in L and removing letters from the beginning and end.

Prove: if L is context free then Substring(L) is also context free.

My intuition is to use the pumping lemma, but in this case we're going from the larger language to the smaller one. Is there a concept of "pumping down" that would work?