Macroeconomics Exam Review 83

Macroeconomics Exam Review 83 - Solutions for Foundations...

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2. (a) Suppose that 𝑋 ( p , 𝑚 ) is not lhc. Then for every neighborhood 𝑆 of ( p , 𝑚 ), there exists ( p , 𝑚 ) 𝑆 such that 𝑋 ( p , 𝑚 ) 𝑇 = . In particular, for every open ball 𝐵 𝑛 ( p , 𝑚 ), there exists a point ( p 𝑛 , 𝑚 𝑛 ) 𝐵 𝑛 ( p , 𝑚 ) such that 𝑋 ( p 𝑛 , 𝑚 𝑛 ) 𝑇 = . (( p 𝑛 , 𝑚 𝑛 )) is the required sequence. (b) By construction, p 𝑛 p < 1 /𝑛 0 which implies that 𝑝 𝑛 𝑖 𝑝 𝑖 for every 𝑖 . Therefore (Exercise 1.202) 𝑝 𝑛 𝑖 ˜ 𝑥 𝑖 𝑝 𝑖 ˜ 𝑥 𝑖 < 𝑚 and 𝑚 𝑛 𝑚 and therefore there exists 𝑁 such that 𝑝 𝑁 𝑖 ˜ 𝑥 𝑖 < 𝑚 𝑁 which implies that ˜ x 𝑋 ( p 𝑁 , 𝑚 𝑁 ) (c) Also by construction 𝑋 ( p 𝑁 , 𝑚 𝑁 ) 𝑇 = which implies 𝑋 ( p 𝑁 , 𝑚 𝑁 ) 𝑇 𝑐 and therefore ˜ x 𝑋 ( p 𝑛 , 𝑚 𝑛 ) = ˜ x / 𝑇 The assumption that 𝑋 ( p , 𝑚 ) is not lhc at ( p , 𝑚 ) implies that ˜ x / 𝑇 , contra-
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