plz do not copy from other answers i have looked them all its not the same question

Use induction over the size of strings to prove that the following NFA over the alphabet sigma = {a, b} recognizes the regular language sigma* ab (that is, it accepts all w element sigma* that end ab). For each string w element sigma* a (i.e., strings that end with an a) all existing computation paths terminate either in state q_1 or in state q_2 (note that we consider only those computation paths that do not "die" in the middle of the computation) For each string w element sigma * ab all existing computation paths terminate either in state q_1 Or in state q_a. For every other string (which is neither in sigma* a nor in sigma* ab, i.e., it belongs to sigma**bb union b union element), all existing computation paths terminate in state q_1. Note that since state q_3 is the only accept state, this statement implies that the language of the above NFA is sigma* ab (see item (2)). Therefore, by proving this statement we also prove that sigma* ab is the language of the above automaton. In the base case you need to show that the statement holds for strings of size 0 and 1. As for the induction step, you need to show that the above induction hypothesis holds for all strings of fixed size n greaterthanorequalto 2, assuming that it holds for all strings of size smaller than n. For example, for each string w element sigma* a you need to show that there exists a computation path that terminates in q_1 and there exists another computation path that terminates in q_2. To do that you can consider the prefix of w of size n - 1, for which the induction hypothesis holds