Given regular language $L$, is $L_1 = \{ w \mid \text{each prefix of } w \text{ of odd length} \in L \}$ regular? I was given a question and don't really know to solve it. 


  Given a regular language $L$, is the following language also regular?
   $$L_1 = \{ w \mid \text{each prefix of } w \text{ of odd length is in $L$} \}.$$


I think that $L_1$ should be regular, but I don't have a clue how to prove it.

Thank you for any input.
Try constructing an NFA for $\hat L:=\{w|$ there is an odd length prefix of $w$ that is $\notin L\}$, given a DFA for $\overline L$, and then show $L_1=\overline {\hat L} $.

If I'm not mistaken, this should work out.