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For subsets X, Y ⊆ R, we define the distance from X to Y as the...

For subsets X, Y ⊆ R, we define the distance from X to Y as the infimum d(X, Y ) := inf D(X, Y ), where

D(X, Y ) := ?{ |x − y| : x ∈ X, y ∈ Y ?}.

Suppose X and Y are sequentially compact.

Prove there exist x ∈ X and y ∈ Y such that |x−y| = d(X,Y).

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