I am trying to show that every deterministic complete finite automaton that recognizes the language $a^*b+b^*$ for $\Sigma = \{a,b\}$ contains a state $q$ such that no accept state can be reached from q on a directed path.
I don't know how to present a diagram here so I use the following link to illustrate the DFA for 'what I think is a good diagram'. Here, I just think the diagram on this page lacks the consideration of an "empty" input so the initial node should be an accept state as well.
However, I am not sure what am I missing in order to prove the question above? Any hints appreciated.
To say that a DFA has such a state is to say that there exists a string $w$ such that no string $wx$ is accepted for any $x\in \Sigma^*$. So we need to find some string that is not the prefix of any accepted word.
Hint: $abb$ is such a string.