Claim:
any two positive integers are equal
Proof:
Let $A(n)$ be statement:
if $a$ and $b$ are any two positive integers such that $\max(a,b)=n$ then $a=b$
Suppose $A(r)$ is true. Let $a$ and $b$ be any two positive integers such that $\max(a,b)=r+1$. Consider the two integers $p=a-1$ and $q=b-1$: then $\max(p,q)=r$. Hence $p=q$, for we are assuming $A(r)$ to be true. It follows that $a=b$; hence $A(r+1)$ is true. $A(1)$ is obviously true, for $max(a,b)=1$ implies $a=b=1$. Therefore by mathematical induction, $A(n)$ is true for every $n$.
Now if $a$ and $b$ are any two positive integers whatsoever, denote $\max(a,b)$ by $r$. Since $A(n)$ has been shown to be true for every $n$, in particular $A(r)$ is true. Hence $a=b$.