Prove that a language is regular if it is accepted by a DFA with more than one intial state
There are many ways of showing that DFAs with multiple initial states generate regular languages. Here are some:


You can prove using Nerode's theorem that for any DFA, the set of words taking the DFA from state $q_1$ to state $q_2$ is regular.
Using "dynamic programming", you can construct a regular expression for the set of all words taking a DFA from state $q_1$ to state $q_2$.
Using $\epsilon$ transitions from a new initial state, you can construct an NFA equivalent to your DFA.


NFAs with multiple initial states are in some sense more natural than NFAs with one initial state. Indeed, the power set construction, which constructs a DFA equivalent to a given NFA, works without any changes for this more general class of NFAs. Moreover, this more liberal definition allows you to prove that regular languages are closed under reversal by simply "reversing all arrows".