Let $(X, p)$ be a compact space with metric $p$. Let $H(X) = \{A \subset X | A=[A]\}$, $p'= max_{a \in A} min_{b \in B} (p(a, b)) + max_{b \in B} min_{a \in A} ((a, b))$. Prove that $(H(X), p')$ is complete metric space.
I've already understood how to prove that $p'$ is metric. But I have question about how to prove its completeness. I think I should use the theorem about completeness of a metric space (A metric space X is complete if and only if every decreasing sequence of non-empty closed subsets of X, with diameters tending to 0, has a non-empty intersection)