= Solution
The <Hölder interpolation inequality> says that for $u\in C^{l,\alpha}(B_R(x_0))$ and every $\varepsilon>0$,
$$
R^l\|D^lu\|_{C^0(B_R)}
\leq\varepsilon R^{l+\alpha}[D^lu]_{C^{0,\alpha}(B_R)}
+C(n,l,\alpha,\varepsilon)\|u\|_{C^0(B_R)}.
$$
More generally, each lower derivative norm can be bounded by an arbitrarily small multiple of the top $C^{l,\alpha}$ seminorm plus a constant multiple of the $C^0$ norm. The powers of $R$ make the inequality invariant under <scaling symmetry of a partial differential equation>[scaling] of the ball.
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