Suppose we want to prove a predicate in the $$HyperReal$$ Numbers,by (weak)induction,for example,for all: $$x\in\mathbb H,x^2>=0$$ Would we Proceed as follows?: $$P(dx)= dx^2>0,$$ $$P(x)=x^2>=0,$$ $$P(x+dx)=x^2+dx^2+2xdx>=0,$$ and since $x^2 ,dx^2$ and $xdx$ are all positive,then for all Hyperreal numbers this predicate holds? If wrong,What is wrong with that proof ? What kind of defintions am i missing? If right , then the can I use the transfer principle to make it true in the real numbers too?