= Solution
For a smooth approximation, the <fundamental theorem of calculus> and <Cauchy-Schwarz inequality> give
$$
|u_n(x)-u_n(y)|\le\int_{[x,y]}|u_n'|\le|x-y|^{1/2}\|u_n'\|_{L^2(I)}.
$$
Let $E\subset I$ be one full-measure set on which $u_n\to u$. Passing to the limit for all $x,y\in E$ gives
$$
\boxed{\mathop{\mathrm{ess\,sup}}_{x\ne y}\frac{|u(x)-u(y)|}{|x-y|^{1/2}}\le\|u'\|_2\le\|u\|_{H^1(I)}.}
$$
The full-measure set $E$ is dense, and the estimate extends its representative uniquely to a $1/2$-<Hölder continuous function> on $\overline I$. Its classical <Hölder seminorm> satisfies the same bound. Changing the original representative on a null set does not affect the <essential supremum> in the assertion.
Back to article page
