Consider what happens when an FA is built for an infinite language over the one-letter alphabet S = {a}. When the input is a string of a’s that is longer than the number of states, the path it traces must take the form of some initial sequence of edges followed by a circuit.
Since all the words in the language accepted by the machine are strings of a’s, all the long words accepted by this FA follow the same path up to the circuit and then around and around as in the picture below: Some of the states leading up to the circuit may be final states and some of the states in the circuit may be final states. This means that by placing + signs judiciously along the path to the circuit we can make the machine accept any finite set of words S1. While going around the circuit the first time the FA can accept another finite set of words S2. If the length of the circuit is n all words of the form an times a word in S2 will also be accepted on the second go-round of the circuit.