= Capacity preservation under uniform limits
Suppose diffeomorphisms preserving capacities of <solid ellipsoids> converge uniformly on compact sets to a diffeomorphism $\phi$. For $0<a<1$, uniform closeness and <degree of a continuous mapping> give $\phi_j(aE)\subseteq\phi(E)\subseteq\phi_j(a^{-1}E)$. The capacity axioms squeeze $c(\phi(E))$ between $a^2c(E)$ and $a^{-2}c(E)$. Letting $a$ tend to one proves equality. Rescaling a smooth $\phi$ near a point and repeating the argument also proves capacity preservation by its derivative.
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