Triangle inequality for pretentious distance

For multiplicative functions $f_1,f_2,g_1,g_2$ bounded in modulus by one, the pretentious distance satisfiesHere products mean pointwise products of arithmetic functions. For complex numbers $z,w$ of modulus at most one, write $a=\Re z$ and $b=\Re w$. The bounds $|\Im z|\le \sqrt{2(1-a)}$ and $|\Im w|\le \sqrt{2(1-b)}$ giveApply this with $z=f_1(p)\overline{g_1(p)}$ and $w=f_2(p)\overline{g_2(p)}$ at each prime $p\le x$, divide by $p$, and use the triangle inequality for the resulting finite Euclidean norm. This proves the displayed product inequality.^[ref]