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Macroeconomics Exam Review 42

# Macroeconomics Exam Review 42 - Solutions for Foundations...

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1.207 The present value of the 𝑛 payments is the 𝑛 th partial sum of the geometric series 𝑥 + 𝛽𝑥 + 𝛽 2 𝑥 + 𝛽 3 𝑥 + . . . which (using (1.31)) is given by Present value = 𝑠 𝑛 = 𝑥 𝛽 𝑛 𝑥 1 𝛽 1.208 By Exercise 1.93, there exists an open set 𝑇 𝑆 1 such that 𝑇 𝑆 2 = . For every x 𝑆 1 , there exists an open ball 𝐵 ( x ) such that 𝐵 ( x ) 𝑇 and therefore 𝐵 ( x ) 𝑆 2 = . The collection { 𝐵 ( x ) } of open balls is an open cover for 𝑆 1 . Since 𝑆 1 is compact there exists a finite subcover, that is there exists points x 1 , x 2 , . . . , x 𝑛 in 𝑆 1 such that 𝑆 1 𝑛 𝑖 =1 𝐵 ( x 𝑖 ) Furthermore, for every x 𝑖 , there exists 𝑟 𝑛 such that 𝐵 ( x 𝑖 ) = x 𝑖 + 𝑟 𝑛 𝐵 where 𝐵 is the unit ball. Let 𝑟 = min 𝑟 𝑛 . 𝑈 = 𝑟𝐵 is the required neighborhood. 1.209 Clearly 𝑋 × 𝑌 is a normed linear space. To show that it is complete, let ( z 𝑛 ) be a Cauchy sequence in 𝑋 × 𝑌 where z 𝑛 = ( x 𝑛 , y 𝑛 ). For every 𝜖 >
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