The claim in the title is described here, along with a non constructive proof.
Here is the claim:
For languages $A$ and $B$ define $A \div B = \{x \in \Sigma^{\ast} : xy \in A \text{ for all } y \in B\}$. If $R$ is regular and $L$ is any language, is it always the case that $R\div L$ is regular?
Is there a constructive proof for this claim? Hence, is there a deterministic way to generate a DFA that accepts $R\div L$, given a DFA that accepts $R$?