vectors inequality

i cant do part b

Consider $|a+b|^2 = (a+b) \cdot (a+b)$.

i have

The theorem is false, now I come to think about it. (Let a={1,0}, b={0,100}.) I suspect there's meant to be a plus sign between |a| and |b|, in which case my hint is correct.
ETA: And in which case, you've proved the triangle inequality as required.