1. The problem statement, all variables and given/known data From Spivak's Calculus Chapter 1: "Suppose that [tex]y_1[/tex] and [tex]y_2[/tex] are not both [tex]0[/tex], and that there is no number λ such that [tex]x_1 =[/tex] λ[tex]y_1[/tex] and [tex]x_2 =[/tex] λ[tex]y_2[/tex]." Then [tex]0[/tex]<(λ[tex]y_1 - x_1)^2 + ([/tex]λ[tex]y_2 - x_2)^2[/tex]. Using problem 18 (which involved proofs related to inequalities like [tex]x^2 + xy + y^2[/tex]), complete the proof of the Schwarz Inequality. 2. Relevant equations None strike me. 3. The attempt at a solution The thing that's really bothering me about this is that the problem I've given is just part a) of the problem. In part d) I am asked to "Deduce...that equality holds only when [tex]y_1 = y_2 = 0[/tex] or when there is a number λ [tex]\geq 0[/tex] such that [tex]x_1 =[/tex] λ[tex]y_1[/tex] and [tex]x_2 =[/tex] λ[tex]y_2[/tex]. Well, in a) he asked me to assume that both of those things were not true to start my proof. Doesn't this mean that, starting with those conditions, one cannot prove that equality is possible, and thus one can't prove the entirety of the Schwarz inequality (as in, the less than or equal to part)?