The Cauchy–Schwarz inequality

For real numbers $a_1,\dots ,a_n$ and $b_1,\dots ,b_n$, show that
$${\left(\sum _{i=1}^n{a}_i{b}_i\right)}^2\le \left(\sum _{i=1}^n{a}_i^2\right)\left(\sum _{i=1}^n{b}_i^2\right),$$
and say exactly when equality holds.

Proofs worth comparing: the discriminant of a quadratic, Lagrange's identity, and a picture about projecting one vector onto another.