What's wrong in this inductive proof? Let $a$ be any positive integer. If $x$ and $y$ are positive integers such that $max(x, y) = a$, then we know that $x = y$.
I'm pretty sure this is a false claim, and I need to find what is wrong with the induction proof.
For the base case, we let $a = 1$. Here, if $x$ and $y$ are positive integers such that $max(x, y) = 1$, then $x$ and $y$ must both be $1$.
In the inductive case, we let $k$ be some arbitrary positive integer. The induction hypothesis assumes that if $x$ and $y$ are positive integers such that $max(x, y) = k$, then $x = y$ for some positive integer $k$.
Consider the case where $a = k + 1$ and let $x'$ and $y'$ be two positive integers such that $max(x', y') = k + 1$. Now we have $max(x' - 1, y' - 1) = k$, which, by the induction hypothesis, implies that $x' - 1 = y' - 1$, and therefore that $x' = y'$. Thus, we have proven the claim.
I can identify just one counterexample: if $a = 20, x = 10, y = 20$. Then, $max(10, 20) = 20$, but $20 \neq 10$.

The problem is that $x'-1$ and $y'-1$ might not both be positive integers (since one of $x',y'$ might be 1), so the induction hypothesis doesn't apply.
