Pf: Assume and Now assume So, By def By the triangle ineq which means since we have since this is a contradiction.
I think there's a somewhat easier way. Suppose that for all . Suppose . Then we know there exists such that implies . This is true because such an implication must hold for any and if then , so this is the particular case where . So this means for . Adding to each part of the equality, we get for , . That means after some point, will be negative! And there arises your contradiction.