How can I prove the Cauchy– Schwarz inequality for two complex numbers? $$z_1=x_1+iy_1$$ $$z_2=x_2+iy_2$$
I can prove the triangle inequality for two complex numbers: $$|z_1+z_2|\le |z_1|+|z_2|.$$ But I cannot prove the Cauchy–Schwarz inequality: $$|z_1\cdot z_2|^2\le |z_2|^2|z_2|^2.$$
In my calculations, I always find the two expressions to be equal.
$a_1, a_2, \ldots, a_n \in \mathbb{C}$ and $b_1, b_2, \ldots, b_n \in \mathbb{C}$: when the $n=1$ $$|\sum_{j=1}^1 a_j \overline{b_j}|^2 \leq \sum_{j=1}^1 |a_j|^2 \sum_{j=1}^1 |b_j|^2$$