fk+ ( / / )( / ) + ( / )( / ) + ( / ) + proving the theorem as second examplewe establish the following theorem theorem if > then  = ( + )( + proof the proof is by induction for the basisit is readily seen that the theorem is true when for the inductive hypothesisassume that the theorem is true for < < we will establish thatunder this assumptionthe theorem is true for we have + = ( ) = applying the inductive hypothesiswe obtain + ( )( ( ) ( ( ( = ( )( )( ( thusn+ = ( )[( ][ ( proving the theorem proof by counterexample the statement fk < is false the easiest way to prove this is to compute proof by contradiction proof by contradiction proceeds by assuming that the theorem is false and showing that this assumption implies that some known property is falseand hence the original assumption was erroneous classic example is the proof that there is an infinite number of primes to prove thiswe assume that the theorem is falseso that there is some largest prime pk let pk be all the primes in order and consider