Find the flaw in this proof that $1$ is the greatest natural number

Question

I think the flaw is in assuming that $N^{2} \in \mathbb{N}$, but I don't know.

Answer 1

The proof is logically correct, and leads to a contradiction (we know that $1<2$). Thus it is a valid proof that there is no greatest natural number.

Answer 2

The flaw is in assuming there is a largest natural number. Assuming that, one can also show that number is $2$, or $17$, or $-3.5$, all at the same time. Ex falso quodlibet.