Proving a DFA recognizes a language using induction
The induction you probably want is to show that a string $w$ ends in state $q_i$ iff it satisfies the property associated to that state.

The basis then is that the empty string, which must end in $q_0$, satisfies the property there.

You should be more precise in some of your properties. In $q_4$, the string ends in 01 as you state, but has never seen any of the special strings, in particular it did not end in 101.