proposition consider the fibonacci function ( )which is defined such that ( ( and (nf( ( for (see section we claim that ( justificationwe will show our claim is correct by induction base cases( < ( and ( induction step( suppose our claim is true for all consider (nsince (nf( ( moreoversince both and are less than nwe can apply the inductive assumption (sometimes called the "inductive hypothesis"to imply that ( - - since - - - - - let us do another inductive argumentthis time for fact we have seen before proposition (which is the same as proposition ii= ( justificationwe will justify this equality by induction base casen trivialfor ( )/ if induction stepn > assume the claim is true for consider - = = by the induction hypothesisthen ni= ( ) which we can simplify as ( ) ( we may sometimes feel overwhelmed by the task of justifying something true for all > we should rememberhoweverthe concreteness of the inductive technique it shows thatfor any particular nthere is finite step-by-step sequence of implications that starts with something true and leads to the truth about in shortthe inductive argument is template for building sequence of direct justifications