How to prove regular languages are closed under left quotient? $L$ is a regular language over the alphabet $\Sigma = \{a,b\}$. The left quotient of $L$ regarding $w \in \Sigma^*$ is the language 
$$w^{-1} L := \{v \mid wv \in L\}$$

How can I prove that $w^{-1}L$ is regular?
Assume $M$ is a deterministic finite state machine accepting $L$. Feed the word $w$ into $M$, which will land you in some state $q$. Construct a new machine $M&#39;$ which is the same as $M$ but has start state $q$. I claim that $M&#39;$ accepts $w^{-1}L$. 

Now prove it.