A compact set and a disjoint closed set have a positive norm-distance gap

Form the set of all admissible pairs $T=\{(a,r)\in K\times (0,\mathrm{\infty }):B(a,3r)\cap C=\mathrm{\varnothing }\}$ and the family $\mathcal{U}=\{K\cap B(a,r):(a,r)\in T\}$. For each fixed $k\in K$, the open complement of $C$ contains $k$, so some $s>0$ has $B(k,s)\cap C=\mathrm{\varnothing }$. Then $(k,s\mathrm{/}3)\in T$ and $k\in K\cap B(k,s\mathrm{/}3)$. Thus $\mathcal{U}$ covers $K$ without selecting radii for all $k$ simultaneously.

Each $V=K\cap B(a,r)$ in this family is open for the restricted metric on $K$. Indeed, if $k\in V$, then $r-\mathrm{\parallel }k-a\mathrm{\parallel }>0$, and every $y\in K$ with $\mathrm{\parallel }y-k\mathrm{\parallel }<r-\mathrm{\parallel }k-a\mathrm{\parallel }$ satisfies $\mathrm{\parallel }y-a\mathrm{\parallel }\le \mathrm{\parallel }y-k\mathrm{\parallel }+\mathrm{\parallel }k-a\mathrm{\parallel }<r$, so is in $V$. Thus $\mathcal{U}$ is an intrinsic open cover of $K$.

Compactness gives a finite subcover $V_0,\dots ,V_{n-1}$ with $n\ge 1$, since $K\mathrm{\ne }\mathrm{\varnothing }$. For each index $j<n$ define $W_j=\{(a,r)\in T:V_j=K\cap B(a,r)\}$. Each $W_j$ is nonempty by the definition of $\mathcal{U}$. Applying finite choice to the function $j\mapsto W_j$ supplies pairs $(a_j,r_j)\in W_j$ for these finitely many indices. Repeated $V_j$ or $W_j$ cause no problem: a choice function on the set of values can be evaluated at each $W_j$.

The finite list $r_0,\dots ,r_{n-1}$ consists of positive reals, so its minimum $\delta$ exists and is positive, since it equals one of those reals.

For any $k\in K$ choose an index $j<n$ with $k\in V_j$, possible because the finite family covers $K$. For every $c\in C$, admissibility gives $\mathrm{\parallel }c-a_j\mathrm{\parallel }\ge 3r_j$, whereas $\mathrm{\parallel }k-a_j\mathrm{\parallel }<r_j$. Hence $\mathrm{\parallel }c-k\mathrm{\parallel }\ge \mathrm{\parallel }c-a_j\mathrm{\parallel }-\mathrm{\parallel }k-a_j\mathrm{\parallel }>2r_j\ge \delta$. This proves the uniform bound for all $k,c$.