Prove that $\texttt{prefix}(L)$ is regular

Given that $ L = \lbrace 0^n1^n : n \geq 0\rbrace$ is a non-regular context-free language, prove that $ \texttt{prefix}(L)$ is regular.

So far I have provided that the grammar to produce this language is: $ S \rightarrow 0S1 \thinspace | \thinspace \epsilon$

Would you go about proving $ \texttt{prefix}(L)$ is regular just like you would any language, proving that $ \Sigma^\star$ = $ \texttt{prefix}(L)$ , or by induction on the length of the words in $ \texttt{prefix}(L)$ .