Note: that this is not precisely the same as this question that has been asked before.
Let $n$ be the largest integer. Then $n>1$. Now $n^2$ is also an integer, and $n^2>n.1=n$. So $n^2>n$. which contradicts $n$ being the largest integer. Therefore our initial assumption is false, and 1 is the largest integer, as claimed.
The problem is that the proof assumes that there is a largest integer. That assumption is false.