
First of all, mathematical induction requires a countable set, so this inductive proof works, the best you can do is the rational numbers, though you can just add a set of measure zero to them, this does not work for all real numbers though, since they are uncountable. Essentially the proof works on a set of cardinality aleph null, but not aleph one. This is also assuming that you are redefining division by zero to be legal. OK. What are you defining 1/0 to be? Whatever you define it to be had better be equal to one when multiplied by zero, since if X=1/0, then X*0=1. Say you choose the extended real number system, putting a point at infinity and negative infinity. OK. 1/0=infinity. So does 2/0. Then 0*infinity=1 and 0*infinity=2, implying that 1=2. Read Goedel for further fun with these implications. 
