Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 31J / c

Fix sample points $(x_i,z_i)$, $1\le i\le n$. A labeling by $h(x,z)=f(x)g(z)$ is the coordinatewise product of one labeling that $\mathcal{F}$ realizes on $x_1,\dots ,x_n$ and one that $\mathcal{G}$ realizes on $z_1,\dots ,z_n$. There are at most the product of the two numbers of choices, so

For fixed $x_1,\dots ,x_n$, the vectoris again the coordinatewise product of a labeling realized by $\mathcal{F}$ and one realized by $\mathcal{G}$, now on the same sample. Hence there are at mostpossible products. Maximizing over samples proves the claimed inequality.