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

Let be the largest integer. Then since is an integer we must have . On the other hand, since is also an integer we must have from which it follows that . Thus, since and we must have . Thus is the largest integer.

I said that the problem was that if is the largest integer, then and . There is no or . So and which contradicts the Trichotomy law.

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

Am I correct?

Thanks