Macroeconomics Exam Review 59

Macroeconomics Exam Review 59 - c 2001 Michael Carter All...

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? ± and therefore ² + ³ ´ ( ± ). Hence, ´ ( ± ) is closed under addition and scalar multiplication; it is a subspace of the linear space µ ( ± ). We conclude that ´ ( ± ) is a normed linear space. 3. To show that ´ ( ± ) is complete, assume that ( ² ? ) is a Cauchy sequence in ´ ( ± ). For every ? ± ² ? ( ? ) ² ± ( ? ) ∣≤∥ ² ? ² ± ∥→ 0 Therefore, for ? ± , ² ? ( ? ) is a Cauchy sequence of real numbers. Since is complete, this sequence converges. De±ne the function ² ( ? ) = lim ? →∞ ² ? ( ? ) We need to show ∙∥ ² ? ² 0and ² ´ ( ± ) ( ² ? ) is a Cauchy sequence. For given ¶> 0, choose · such that ² ? ² ± <¶/ 2 for very ¸,¹ · . For any ? ± and ¹ · , ² ? ( ? ) ² ( ? ) ∣≤∣ ² ? ( ? ) ² ± ( ? ) + ² ± ( ? ) ² ( ? ) ≤∥ ² ? ² ± + ² ± ( ? ) ² ( ? ) By suitable choice of ¸ (which may depend upon ? ), each term on the right can be made smaller than ¶/ 2 and therefore ² ? ( ? ) ² ( ? ) for every ? ± and ¹ · . ² ? ² =sup ² ³ ² ? ( ? ) ² ( ? ) ∣≤ Finally, this implies ² = lim ? →∞ ² ? . Therefore
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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