Lower semicontinuity of a Hölder seminorm ($f_j\to f\text{ uniformly}⟹[f{]}_{\mu }\le {\mathrm{liminf}}_j[f_j{]}_{\mu }$)

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.