Claim1 : If L(M) is infinite, then there must exist a cycle in the directed graph.

Proof : Since L(M) is infinite, according to the previous theorem, there exists a string with where n is the number of states in the DFA M . Since the length of the accepted string w is greater than the number of states, there must exist a repeated state in the path from q₀ to the final state while processing the string w. His repetition of (at least one) state in the path implies the existence of a cycle.

Claim 2: If there is a cycle in the directed graph (for the DFA M ), then L(M) must be infinite.

Proof : We know that all states are reachable from the start state q₀. Also, there can not be any cycle involving “useless” states, because these have already been removed.

Hence if there exists a cycle, there must be a path from the start state q₀ to one of the states involved in the cycle and, also, there must be a path from on e of the states involved in the cycle to an accepting state. The situation is depicted in the following figure.