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On a fixed smooth bounded domain, zero-boundary solutions with forcing in a bounded subset of $C^{0,\mu }$ are uniformly bounded in $C^{2,\mu }$ by a maximum-principle barrier and the global Schauder estimate. The Arzela-Ascoli theorem gives convergence in $C^2$. A closed Hölder-norm constraint passes to the limit by lower semicontinuity of a Hölder seminorm, so the constrained solution set is compact rather than merely relatively compact.

If $f_j\to f$ uniformly and the Hölder seminorms are uniformly bounded, each fixed difference quotient converges. Taking the supremum over point pairs gives $[f{]}_{\mu }\le {\mathrm{liminf}}_j[f_j{]}_{\mu }$. The supremum norms converge, so the full Hölder norm has the same lower semicontinuity of a Hölder seminorm.