geometric model

I don't know what you mean by geometric models.

(1) follows from (2) - set b=1.

2) a and b must have the same sign and both are nonzero
multiply by ab to get a^2 + b^2 >= 2ab
subtract 2ab to get a^2 - 2ab + b^2 >= 0
the left side is a square, so we have (a - b)^2 >= 0 which is always true.