supremum and infimum - Why is the lagrange dual function concave

In a book I'm reading (Convex Optimization by Boyd and Vandenberghe) it says I'm struggling to understand the last sentence. Why can one conclude concavity from having a pointwise infimum of a fam

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Why does infimum = supremum for this dual function simplification

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2. The infimum and supremum in I for C = R 2 + .