Proving regular languages

Question

I am given the language L = {a,b}* and a/L = { w ∈ {a,b}* | aw ∈ L }. I am trying to prove that that if L is regular so is a/L. My approach so far is the prove that L is regular (using pumping lemma) and since a/L is a subset of L, then a/L must also be regular. I am having trouble doing the pumping lemma on L? What is an example string S I can use?

Answer 1

HINT: The pumping lemma cannot be used to prove that a language is regular: it can only be used to show that a language is not regular. I suggest starting with a DFA that accepts $L$ and showing how to modify it to get an DFA that accepts $a/L$; the required modification is very simple.