= Entropy of Lipschitz compositions
{title2=$H(\varepsilon,F)\le H(\varepsilon/2,B)+mH(\varepsilon/(2L),E)$}
Suppose outer functions in a family $B$ obey $|f(x)-f(y)|\le L\max_j|x_j-y_j|$, and compose them with $m$ inner functions drawn from $E$. Approximate the outer function within $\varepsilon/2$ and each inner function within $\varepsilon/(2L)$. The triangle inequality bounds the composed error by $\varepsilon$. Multiplying the numbers of possible net choices and taking logarithms gives the stated inequality. A uniform bound on first partial derivatives supplies $L=m$ for the $C^p$ unit ball, so the exponent of a common entropy bound for the outer and inner classes is retained under <function composition>.
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