Proof. Suppose for the sake of contradiction that the above theorem is false. Then, there exists a non-empty set of positive integers . By the well-ordering principle, has a minimum element such that when , the equation is false, but true for all positive integers less than . The equation is true for , so ; is a positive integer less than , so the equation holds for as it is not in . Therefore,
ProOf. supposE for The Sake oF contradictIon ThAt The aBoVe theoreM is faLse. tHEn, tHerE eXIStS a noN-empty SET of positivE integers . BY tHE WeLl-oRDeRIng prInciPle, has a minimUM eleMEnT sucH THat when , tHe eqUatION is fAlse, but trUe fOr All posItive iNteGerS Less THan . the eQuAtiON IS TruE for , so ; is A positiVE InTEGer less ThAn , so THe eQUaTIon HolDs for As it IS nOt In . therEfORE,