I need to prove that $L(f(M)) = L(M)\cup \{\varepsilon\}$where $M$ is a DFA and $f$ is the function $f(M) := (Q\cup \{q_f\}, \Sigma, \delta', q_f, F\cup\{q_f\})$ and $q_f$ is a new state not in $Q$ and $\delta'(q,a) =       \begin{cases}        \delta (q,a)  & \text{if }q\in Q\\        \delta (q_0,a) & \text{if }q= q_f.      \end{cases}$I'm assuming I need to use induction but I'm not sure how to go about it
How do you prove two languages are equivalent using the definition of acceptance?
formal languages;regular languages;finite automata;proof techniques;induction