Assume the first expression is true for all [itex]x_1, x_2, \cdots, x_n[/itex]. This means you can hand-pick a particular [itex]x_1, x_2, \cdots, x_n[/itex] that makes the desired relation fall out.
Almost. You get norm(r)^2 >= norm(r)^4, and thus 1 >= norm(r)^2.
To be completely rigorous you also have to handle the special case where the division in the final step is not valid. This case is trivial, so to speak.