Given that $L_2$ is regular and infinite and $L_1 \cdot L_2$ is regular, then $L_1$ is also regular.I need some help on getting started on proving this is the case.My intuition is that if $L_1 \cdot L_2$ is regular there exists a DFA for it, in which case you can just remove the states that correspond to the $L_2$ part of the DFA and you are left with a DFA that recognizes $L_1$.
If both the concatenation of two languages and the second half are regular, is the first too?
formal languages;regular languages;closure properties