Unformatted text preview: 1.217 If int is empty, it is trivially convex. Therefore, assume int ∕ = ∅ and let x , y ∈ int . We must show that z = x + (1 − ) y ∈ int . Since x , y ∈ int , there exists some > 0 such that the open balls ( x , ) and ( y , ) are both contained in int . Let w be any vector with ∥ w ∥ < . The point z + w = ( x + w ) + (1 − )( y + w ) ∈ since x + w ∈ ( x , ) ⊂ and y + w ∈ ( y , ) ⊂ and is convex. Hence z is an interior point of . Similarly, if is empty, it is trivially convex. Therefore, assume ∕ = ∅ and let x , y ∈ . Choose some . We must show that = x + (1 − ) y ∈ . There exist sequences ( x ) and ( y ) in which converge to x and y respectively (Exercise 1.105). Since is convex, the sequence ( x + (1 − ) y ) lies in and moreover converges to...
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- Fall '10
- Macroeconomics, Existence, All rights reserved, ball, Topological space