Answer
The inductive step fails when x and/or y equals 1
Work Step by Step
Let x equals 1. The inductive hypothesis thus fails because the result only talks about both the inputs to the max function being positive integers.
But when we say max(x-1, y-1) our first argument (i.e. x-1) is not positive (in fact it is 0).
So we can not apply the result and hence our proof is faulty.
