Macroeconomics Exam Review 160

Macroeconomics Exam Review 160 - y ∈ Since y ∈ y must...

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Unformatted text preview: y ∈ Since y / ∈ , y must be a boundary point of . By the previous exercise, there exists a supporting hyperplane at y , that is there exists a continuous linear functional ∈ ∗ such that ( y ) ≤ ( x ) for every x ∈ 3.185 1. ( ) ⊆ ℜ . 2. ( ) is convex and hence an interval (Exercise 1.160. 3. ( ) is open in ℜ (Proposition 3.2). 3.186 is nonempty and convex and / ∈ . (Otherwise, there exists x ∈ and y ∈ such that = y + ( − x ) which implies that x = y contradicting the assumption that ∩ = ∅ .) Thus there exists a continuous linear functional ∈ ∗ such that ( y − x ) ≥ ( ) = 0 for every x ∈ , y ∈ so that ( x ) ≤ ( y ) for every x ∈ , y ∈ Let = sup x ∈ ( x ). Then ( x ) ≤ ≤ ( y ) for every x ∈ , y ∈ By Exercise 3.185, (int ) is an open interval in (...
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