Do all DFA's containing an "accepting path containing a cycle" accept infinite languages?
If the DFA contains a path from the start node to an accepting state that contains a cycle, then the DFA is an infinite language.  The proof is easy: you can go around the cycle as many times as you like.  Depending on how many times you go around the cycle, you get a different word accepted by the DFA; the more times around the cycle, the longer the word (so it has to be different from any word where you went around the cycle a different number of times). There are infinitely many possibilities for how many times you can choose to go around the cycle, so infinitely many words accepted by the DFA.