Let $L, M \subseteq \Sigma^*$ be regular languages. I need to prove that $N = \{x \in \Sigma^* \; | \; \exists y \in L : xy \in M\}$ is as well a regular language.
My favored approach is to find a finite state machine that recognizes that language. I thought about the FSM that recognizes M and $\lambda$-transitions to the FSM of $L$, but this didn't help me on it. I don't know how to integrate the recognition of $L$ to each step of $M$ and am stuck.
Could you please help me with that?
Thanks in advance!