Show that every finite subset of $\{0,1\}^*$ is recursive

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How can I show that every finite subset of $\{0,1\}^*$ is recursive ?

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Every finite subset of $\left \{ 0,1 \right \}^*$ like $L$ is a regular language. It is easy to construct a DFA $M$ such that $L(M)=L$. Now Let $M'$ be a turing machine such that works like $M$ and in every transition head goes right on tape.