Let $L$ be a regular language. Prove that $L_1$, the language created by removing all characters in odd places in all words of $L$, is regular

Question

Let $L$ be a regular language. Prove that $L_1$, the language created by removing all characters in odd places in all words of $L$, is regular.

Completely stuck on this one. I tried building DFA,NFA,$\epsilon$-NFA, product construction of automata, but I'm not sure how to even begin.

Answer 1

Note that $L_1$ is the image of $L$ under the sequential automaton represented below, where $1$ is the empty word and $a$ is any letter of the alphabet.

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Since regular languages are closed under sequential automata, $L_1$ is regular.