Prove that there is no positive integer between 0 and 1
I think this is a weird example chosen to illustrate the version of induction that uses the well ordering of the positive integers. That means you can assume whatever you need about arithmetic, including squaring, and the fact that for real numbers $a < 1$ implies $a^2 < a$.
Then, as the argument says, the square of some (hypothetical) integer between $0$ and $1$ will be less than $a$.