Suppose $(E,d)$ is a complete metric space. Let $\{A_n\}_{n\in\mathbb{N}}$ be a sequence of compact sets in $E$ such that $d(x,A_n)$ converges uniformly in $x$ to some function $d:E\to \mathbb{R}$ as $n\to\infty$, i.e. $d(x,A_n)\rightrightarrows d(x)$ uniformly for $x\in E$. Let $K=d^{-1}(0)$. Is it true that $d(x)=d(x,K)$? (I have proved that K is compact.)
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