# Analysis 1 Midterm - Analysis I section 5 Midterm[100 marks...

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Analysis I, section 5 - Midterm (10/22/2013) [100 marks] Name: Please justify all your answers, unless otherwise instructed. No calculators allowed. Duration: 75 minutes. 1. Provide precise definition for the following objects (4 marks each) :
2. Formulate the following theorems precisely (you do not have to prove them! (6 marks each) ) Comparison Theorem for real sequences Nested interval property Sequential characterization of limits of real functions Extreme Value Theorem
3. Prove the Bolzano-Weierstrass Theorem. You can assume Nested Interval Property. (15 marks)
4. Decide which of the following statements are true and which are false. Prove the true ones and give counterex- amples to the false ones (7 marks each) . If A and B are nonempty bounded subsets of R , then sup( A B ) = max { sup A, sup B } . If { x n } and { y n } are Cauchy with y n 6 = 0 for all