Proof: Two Cases: i.) For a=0: If a=0 and b>a ##\Rightarrow ## a^2=0^2=0. Thus, a^2 < b^2. ii.) For a>0: If a>0 and b>a ## \Rightarrow ## b-a>0 and b+a>0. By closure under addition, (b-a)(b+a)>0. Or, b^2-a^2>0. Or, b^2>a^2. Or, a^2<b^2. My friend said that this proof is wrong. I think he's just telling me that to get me mad because I don't see what I did wrong. I feel stupid because it's seems so obvious. Maybe I just don't have what it takes to work through Spivak .