The infomal pumping lemma principle can be stated as: (for regularity checking similarly can be reduced to CFL as well) If there is a state in Automaton which is self loop and machine pumped it once only to accept the string in the language yet pumped twice will produce a string which is not in language therefore its not regular .Formal definition can be studied from here .
Now,There is always a finite state is a finite automaton and if a infinite language is generated therefore it implies that there is no one to one association b/w the sets. and therefore it implies that there must be two element in the codomain which have same image which can be pumped again .
Similar to statment of PIGENHOLE. which states the if a set has $N$ elements and other set has $M$ elements and there is one to one mapping but $N>M$ which implies one to one mapping is not at all possible.
Footer Note : No such statments can be made for recursion and iterative method therefore can be eliminated.
