Show for any state $q$, string $x$, and input symbol $a$, $\hat\delta(q, ax) = \hat\delta(\delta(q, a), x)$, where $\hat\delta$ is the transitive closure of $\delta$, which is the transition function for some DFA.
I think the best way to proceed is by induction and that the following is the basis step:
Basis: $\hat\delta(q, a) = \hat\delta(\delta(q, a), \epsilon)$
But I am not sure how to proceed to the inductive step as I'm a little unsure how to use induction in these cases: it seems much different from numerical induction. Can someone provide a suggestion?