> Multiplication* an arithmetical operation, defined initially in > terms of repeated addition, usually written a × b, a.b, or ab, > by which the product of two quantities is calculated: to multiply > a by positive integral b is to add a to itself b times. > > i.e. ab = a added to itself b times > > This definition fails proof by induction. > > So what other proofs can be used to prove ab does not equal > a added to itself b times?
I don't follow your argument. Assuming that something "fails proof by induction" , it does not follow that the result is not true. Maybe it can be proved by another method.
 By the way, this is not clearly phrased. Do you mean every proof by induction must fail or some proof by induction must fail? Also, I think you're using the term "proof" differently than mathematicians use it (i.e. a logically correct argument), even aside from the liberal use of "proof of a defintion" (by which I assume you mean something like "a proof that something or other fits the criterion for a definition").