Consider the Fréchet space $\Omega = C(\mathbb R^d)$ of real-valued continuous functions equipped with the seminorms $$\|f\|_D := \sup_{x,y \in D} \left\{ |f(x)|, \tfrac{|f(x)-f(y)|}{|x-y|} \right\}, \qquad \mathrm{for~compact~} D \subseteq \mathbb R^d.$$
That is, $\|f\|_D$ is the larger of the supremum norm and the Lipschitz constant of $f$ over $D$.
Let $D \subseteq \mathbb R^d$ be compact, and let $f \in \Omega$. Consider a sequence $D_n$ of compact sets and a sequence of functions $f_n$ so that:
- $D_n \to D$ in the Hausdorff topology,
- $f_n \to f$ in $\Omega$, and
- There exists a value $h$ so that $\|f_n\|_{D_n} \le h$ for all $n$.
Is it the case that $\|f\|_D \le h$?
