What\'s wrong with the following (incorrect) proof of this (incorrect) theorem?

Question

What's wrong with the following (incorrect) proof of this (incorrect) theorem?

Theorem: If n is a natural number, and p, q are any two natural numbers that have a maximum value of n, then p=q.

Proof: Base case n = 1: p, q are natural numbers so p, q >= 1. If their maximum value is 1, then p=q=1.
Suppose for n = k, we have p, q are any natural numbers with a maximum of k, then p = q.
For n = k+1, suppose p, q have a maximum value of k+1. Then p-1, q-1 have a maximum value of k. By hypothesis, this means p-1=q-1, but adding 1 to both sides gives p=q.

Explanation / Answer

For n = k+1, suppose p, q have a maximum value of k+1. Then p-1, q-1 have a maximum value of k. By hypothesis, this means p-1=q-1, but adding 1 to both sides gives p=q.

It's true if n=1 because if p and q are both natural numbers and their maximum is 1, well the first natural number is 1, so p and q must both be 1, otherwise one would be less than 1 but then it's not a natural number.

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