The following DFA accepts the set of all strings over {0,1} that

The DFA starts at the initial state, looping on '1' to ignore leading ones. A transition on '0' moves to an intermediate state, indicating the last character was a single zero. If another '0' is read, it reaches the accepting state; otherwise, a '1' resets to the start. The final state loops on '0', ensuring any subsequent zeros maintain acceptance. Thus, it accepts strings ending with two consecutive zeros.