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Math 414: Analysis I Homework 2 Due: February 3rd, 2014 Name: The following problems are for additional practice and are not to be turned in: (All problems come from Basic Analysis, Lebl ) Exercises: 1.1.1, 1.1.3, 1.1.4, 1.1.5, 1.2.1, 1.2.2, 1.2.5, 1.2.8, 1.2.10 Turn in the following problems. 1. (a) Show that if x, y are rational numbers, then x + y and xy are rational numbers. (b) Prove that if x is a rational number and y is an irrational number, then x + y is an irrational number. If, in addition, x 6= 0, then show that xy is an irrational number. 2. If 0 ≤ a < b, show that a2 ≤ ab < b2 . Show by example that it does not follow that a2 < ab < b2 . 3. Let a, b ∈ R. Suppose that for every > 0, we have a ≤ b + . Show that a ≤ b. 4. (a) Let S1 := {x ∈ R : x ≥ 0}. Show in detail that S1 has lower bounds, but no upper bounds. Show that inf S1 = 0. (b) Now let S2 := {x ∈ R : x > 0}. Does S2 have lower bounds? upper bounds? Does inf S2 exist? Does sup S2 exist? Prove your answers. 5. Let S be a nonempty subset of R that is bounded below. Prove that inf S = − sup {−s : s ∈ S} . 6. Let S S ⊂ R and suppose that s∗ := sup S belongs to S. If u ∈ / S, show that sup (S {u}) = sup {s∗ , u}. 7. Exercise 1.1.2 from Basic Analysis, Lebl. (Without loss of generality, you can prove just the existence of sup A.) Hint: Use induction and the preceding exercise. 8. Exercise 1.2.7 from Basic Analysis, Lebl. 9. Exercise 1.2.9 from Basic Analysis, Lebl. 1