Help with Cauchy-Schwarz Inequality (when it is an equality) (from Apostol's Calculus)

Question

Apostol tells the reader to verify the last sentence. I've verified the backward implication ($a_kx+b_k=0\implies \text{equality}$). I'm having trouble verifying the forward implication. Could someone show me a proof of the forward implication?

Answer 1

The forward implication is actually false.

Counterexample:

Let $n=1$, with $a_1=0$ and $b_1=9$. Then clearly, both the left and right-hand sides of the inequality equal zero. But there is no real $x$ which will make $0x+9=9=0$ true.

Answer 2

If we have equality then $\mid x\cdot y\mid^2=\mid x\mid^2\cdot \mid y\mid^2$. But lhs$=\mid x\mid^2\cdot \mid y\mid^2\cos^2\theta$. $\therefore \cos^2\theta=1\implies \theta =0$ or $\pi\implies $ the angle between $(a_1,\dots,a_n)$ and $(b_1,\dots,b_n)$ is $0$ or $\pi$. So they are multiples of each other; or one of them is zero.