What is wrong with this proof that 1 is the largest integer? What does it prove?

Let $\displaystyle n $ be the largest integer. Then since $\displaystyle 1 $ is an integer we must have $\displaystyle 1 \leq n $. On the other hand, since $\displaystyle n^{2} $ is also an integer we must have $\displaystyle n^{2} \leq n $ from which it follows that $\displaystyle n \leq 1 $. Thus, since $\displaystyle 1 \leq n $ and $\displaystyle n \leq 1 $ we must have $\displaystyle n = 1 $. Thus $\displaystyle 1 $ is the largest integer.

I said that the problem was that if $\displaystyle 1 $ is the largest integer, then $\displaystyle 1 < n $ and $\displaystyle n^{2} < n $. There is no $\displaystyle \leq $ or $\displaystyle \geq $. So $\displaystyle n < 1 $ and $\displaystyle n > 1 $ which contradicts the Trichotomy law.

The above argument that $\displaystyle 1 $ is the largest integer proves that direct proofs arent the best way to approach a problem?

Am I correct?

Thanks