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PROBLEM SET 2 (DUE IN LECTURE ON SEP 25) (All Theorem and Exercise numbers are references to the textbook by Apostol; for instance “Exercise I 3.5-5” means Exercise 5 in section I 3.5.) Problem 1. Let n be an integer and let x be a real number satisfying n < x < n + 1. Prove that x is not an integer. (You may assume without proof that the sum or difference of integers is an integer.) Problem 2. Do Exercise I 3.12-4. Problem 3. Do Exercise I 4.10-4. (Binomial coefficients are defined at the beginning of section 4.10. You can use Exercise I 4.10-3.) Problem 4. Do Exercise I 4.10-16. Problem 5. Do all parts of Exercise 1.5-9. Problem 6. Do parts (a) and (c) of Exercise 1.5-10. (Hint: Consider the polynomial q(x) = p(x) − 1.) Problem 7. Prove that 0.999... = 1. 1