What is wrong with the following argument, which allegedly shows that any two po

Question

What is wrong with the following argument, which allegedly shows that any two positive integers are equal?
We use induction on n to "prove" that if a and b are positive integers and n=max{a,b}, then a=b.
Basis step (n=1). If a and b are positive integers and 1=max{a,b}, we must have a=b=1.
Inductive step. Assume that if a' and b' are positive integers and n=max{a',b'}, then a'=b'. Suppose that a and b are positive integers and that n+1=max{a,b}. Now n=max{a-1,b-1}. By the inductive hypothesis, a-1=b-1. Therefore, a=b.
Since we have verified the Basis step and the Inductions step, by the Principle of Mathematical Induction, any two positive integers are equal!

Explanation / Answer

Sufficient to show that for any two positive integers, A and B, A = B. Further, it is sufficient to show that for all N > 0, if A and B (positive integers) satisfy (MAX(A, B) = N) then A = B. Proceed by induction. If N = 1, then A and B, being positive integers, must both be 1. So A = B. Assume that the theorem is true for some value k. Take A and B with MAX(A, B) = k+1. Then MAX((A-1), (B-1)) = k. And hence (A-1) = (B-1). Consequently, A = B.

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