Make sure you sort out all the geometry from the proof to be sure we really have a proof here and not just heuristic hand waving. Another approach entirely Let's go back to the notion of angle between two vectors. Along with the dot product, you had a vector cross product in the case of . It gave you a vector that was perpendicular to the original two vectors, say v and w, and its length was precisely . It could also be calculated as the value of the determinant of a strange looking matrix in which the top row had the unit vectors i, j, and k in the coordinate axis directions and the other two rows had the coordinates of v and w respectively.
Identities are mysterious and wonderful.
If you write the above in terms of all the coordinates, then you simply have an algebraic identity. Use the above identity to prove the Cauchy-Schwartz inequality. Now generalize it to n dimensions and get another proof of Proposition 2.