Let $n$ be the largest positive integer. Since $n ≥ 1$, multiplying both sides by $n$ implies that $n^2 ≥ n$.
But since $n$ is the biggest positive integer, it is also true that $n^2 ≤ n$. It follows that $n^2 = n$. Dividing both
sides by $n$ implies that $n = 1$.
The goal is to find the flaw in the reasoning of the proof rather than find a proof that proves it wrong. Here's where I think the problem is:
It was stated that $n^2 ≥ n$ after multiplying both sides of the inequality by $n$. But then because $n$ is the largest possible integer, $n^2 ≤ n$. This is where I think the flaw is, because if $n$ is the single largest integer, then we'd get $n^2 < n$ rather than $n^2 ≤ n$, so writing the latter is incorrect.
Best Answer
In fact, you have given a valid proof of a true theorem.
Theorem. If $n$ is the largest positive integer, then $n=1$.
You would start a proof of this theorem exactly the way you did start: "Let $n$ be the largest positive integer." So your proof is perfectly valid. But it doesn't prove that $1$ is the largest positive integer; that would be a different theorem: There exists a largest positive integer, and it is equal to $1$. To prove that stronger theorem, you'd first have to prove existence, which of course you cannot do.
The theorem statement you did prove is an example of a mathematical statement that is "vacuously true." This means it is true because its hypothesis is always false. If you look at the truth table for the implication $P\implies Q$, you'll see that in all cases in which $P$ is false, the implication is true. So you proved a true (but entirely uninteresting) result!