Explain in detail why the following “proof” is incorrect.
Theorem $1$. All positive integers are equal.
Proof. We show that any two positive integers are equal, from which the result follows. We do this by induction on the maximum of the two numbers. Let $P (n)$ be the statement “if $r$ and $s$ are positive integers and max{$r, s$} = $n$ then $r = s$”.
Clearly $P (1)$ is true. Suppose that $P (n)$ is true and let $r, s$ be positive integers whose maximum is $n + 1$. Then max{$r−1,s−1$} = $n$. By the inductive hypothesis, $r−1=s−1$ and hence $r=s$. Thus $P(n+1)$is true.
The result is now proved by mathematical induction.
