Can't find the mistake in my inductive step

Question

Few days ago I was solving some induction execises, and I tried and solved this one.

Statement

What is wrong with this “proof”?

“Theorem” For every positive integer $n$, if $x$ and $y$ are positive integers with $\max(x, y) = n$, then $x = y$.

Basis Step: Suppose that $n = 1$. If $\max(x, y) = 1$ and $x$ and $y$ are positive integers, we have $x = 1$ and $y = 1$.

Inductive Step: Let $k$ be a positive integer. Assume that whenever $\max(x, y) = k$ and $x$ and $y$ are positive integers, then $x = y$. Now let $\max(x, y) = k + 1$, where $x$ and $y$ are positive integers. Then $\max(x − 1, y − 1) = k$, so by the inductive hypothesis, $x − 1 = y − 1$. It follows that $x = y$, completing the inductive step.

Solution, taken from the original book

The mistake is in applying the inductive hypothesis to look at $\max(x − 1, y − 1)$, because even though $x$ and $y$ are positive integers, $ x − 1$ and $y − 1$ need not be (one or both could be 0)

Now my question

After solving the problem I wrote my own inductive step, assuming the same hypothesis. I did it just for fun, but now, even knowing that my inductive step is wrong, I can't find the mistake. I need to know what is wrong in my inductive step and why.

My inductive step

Inductive Step: Let $k$ be a positive integer. Assume that whenever $\max(x, y) = k$ and $x$ and $y$ are positive integers, then $x = y$. Since $\max(x, y) = k$ and $x$ and $y$ are positive integers with$ x = y$, I add $1$ to both $x$ and $y$. Then $\max(x + 1, y + 1) = k + 1$. It follows that $x + 1 = y + 1$ because $x = y$, completing the inductive step.

Answer 1

The statement you are claiming to prove is the statement

If $\max(x,y)=n$, then $x=y$.

However, the statement you prove is the statement

If $x=y$, then $\max(x,y)=n$

In other words, your inductive step is correct because it proves a correct statement, but not the same statement as you think you are proving.

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If you really wanted to prove your statement, you would have to prove that if $\max(x+1, y+1) = k+1$, then $x+1=y+1$.

Instead, you only proved the statement if $x=y$, then $x+1=y+1$.

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Another way of looking is you only indictively proved the statement for pairs $(x',y')$ which can be written as $(x+1, y+1)$, where $x,y$ are positive integers. So, you didn't prove the statement for $(0,2)$, for example.

Answer 2

Your inductive step is correct, but it doesn't prove the theorem, because you're working backwards:

$x=y\Rightarrow \max(x+1,y+1)=k+1.$

Whereas you should be proving:

$\max(x+1,y+1)=k+1\Rightarrow x+1=y+1$