The Squared-Distance Penalization Limit on a Compact Set

Let $(X,d)$ be a metric space, equipped with the collection of its subsets that are open in $(X,d)$, a topology by Metric Open Sets Form a Topology, and let $K\subseteq X$ be nonempty and compact in $X$. Let $\mathbb{R}$ be the set of real numbers with the addition, multiplication and order $\le$ of its ordered field structure, and write $s-t$ for $s+(-t)$. Equip $X\times X$ with the product metric $d_{X\times X}$ obtained from $d$ and $d$, a metric by claim 1 of The Product Metric is a Metric. Let $u:K\to\mathbb{R}$ be upper semicontinuous on $K$ and let $v:K\to\mathbb{R}$ be lower semicontinuous on $K$.

Let $\psi:K\times K\to\mathbb{R}$ be the function $\psi(x,y)=d(x,y)\,d(x,y)$, whose value we also write $d(x,y)^{2}$.

Then $\psi$ is lower semicontinuous on $K\times K$ with respect to $d_{X\times X}$, satisfies $0\le\psi(x,y)$ for every $(x,y)\in K\times K$, and satisfies $\psi(x,y)=0$ if and only if $x=y$. Consequently the hypotheses of Limits of Penalized Maxima on a Compact Set hold for this $\psi$, and claims 1 to 6 of that theorem hold for the functions