Please prove or disprove the following conjecture: If C is non-compact, closed subset of the metric space X and x is any point in X then there exists a point c belonging to C such that d(x,c) = d(x,C).
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Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Apr 07 '23 at 14:29
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This may help: https://math.stackexchange.com/questions/938367/distance-between-a-point-and-a-closed-set-in-metric-space – Son Gohan Apr 07 '23 at 14:32