Macroeconomics Exam Review 46

# Macroeconomics Exam Review 46 - 1.217 If int is empty it is...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online