Let A be a subset of a complete metric space. Assume that for all ε > 0, there exists a compact set Aε so that ∀x ∈ A, d(x, Aε) < ε. Show that A (close) is compact

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Let A be a subset of a complete metric space. Assume that for all

ε > 0, there exists a compact set Aε so that

x A, d(x, Aε) < ε. Show that A (close) is compact.

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