I'm reading Apostol's Calculus.
It says that due to the Cauchy-Schwarz inequality written as:
$$|\langle A,B\rangle|\leq ||A||\, ||B||$$
Then
$$-1\leq\frac{\langle A,B\rangle}{||A||\, ||B||}\leq 1$$
I am a bit confused at this. I did the following to check that if:
$$A=(a_1,a_2,a_2) \quad B=(b_1, b_2, b_3)$$
Then:
$$\frac{a_1b_1+a_2b_2+a_3b_3}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}}$$
But from here, I can only expand the square roots. I don't know how to proceed.