Lower semicontinuity of a Hölder seminorm
= Lower semicontinuity of a Hölder seminorm
{title2=$f_j\to f\text{ uniformly}\Longrightarrow[f]_\mu\leq\liminf_j[f_j]_\mu$}
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 $[f]_\mu\leq\liminf_j[f_j]_\mu$. The <supremum norms> converge, so the full <Hölder norm> has the same <lower semicontinuity of a Hölder seminorm>.