HW1 for MAT 337 Spring 2021

Due February 9

1 Section 2.2, exercises E, F, G, H, I.

2 Section 2.3, exercises B, C, D, E.

3 Give an example of each of the following, or state that the request is impossible (and explain why).

(a) A set B with inf B ≥ sup B.

(b) A finite set that contains its infimum but not its supremum.

(c) A bounded subset of Q that contains its supremum but not its infimum.

4 Given sets A and B, define A + B = {a + b : a ∈ Aandb ∈ B}. Follow these steps to prove that if A and B are nonempty and bounded above then sup(A + B) = sup A + sup B.

(a) Let s = sup A and t = sup B. Show s + t is an upper bound for A + B.

(b) Now let u be an arbitrary upper bound for A+B, and temporarily fix a ∈ A. Show t ≤ u - a.

(c) Finally, show sup(A + B) = s + t.

5 Decide if the following statements about suprema and infima are true or false. Give a short proof for those that are true. For any that are false, supply an example where the claim in question does not appear to hold.

(a) If A and B are nonempty, bounded, and satisfy A ⊆ B, then sup A ≤ sup B.

(b) If sup A < inf B for sets A and B, then there exists a c ∈ R satisfying a < c < b for all a ∈ A and b ∈ B.

(c) If there exists a c ∈ R satisfying a < c < b for all a ∈ A and b ∈ B, then sup A < inf B.

6 Section 2.4, exercises A, B, C, G.

7 Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are not:

(a) Two real numbers satisfy a < b if and only if a < b + ε for every ε > 0.

(b) Two real numbers satisfy a < b if a < b + ε for every ε > 0.

(c) Two real numbers satisfy a ≤ b if and only if a < b + ε for every ε > 0.2

8 Section 2.5, exercises B, E, H, I.

9 Section 2.6, exercises B, D, E, F, K.

10 Section 2.7, exercises F, G, H, I, J.

11 Section 2.8, exercises I, J.

12 Section 3.1, exercises B, C, G.

13 Section 3.2, exercises F, O, P.

14 Section 3.3, exercises A, B, E.