In chapter 2 of the New Turing Omnibus, the author considers an unknown finite automata with 6 states. Through trial and error, it is deduced that the words 0101, 0100101, 0100100101 are accepted. It is then stated that using the pumping lemma, it can be shown that if 01(001)^n01 is accepted for n=0,1,2,3,4,5 then all strings of the form 01(001)*01 must also be accepted.
I know of the pumping lemma but do not see how the next leap has been made. Why is it sufficient to only test n up (number of states) - 1?