Math 201-B
Introduction to Proofs
Instructor: Alex Roitershtein
Iowa State University
Department of Mathematics
Fall 2015
Hw #7 (practice test)
Due date: December 9, 2015
1. [20 points]
(a) Given a set A ⊂ R define a set −A (negative A) by setting
−A := {x ∈ R : −x ∈ A}.
Show that for a bounded set A we have sup A = − inf(−A) and inf A = − sup(−A).
(b) Given two sets A ⊂ R and B ⊂ R define a set A + B by setting
A + B = {x ∈ R : ∃ a ∈ A, b ∈ B s. t. x = a + b}.
Is it true that for bounded sets of reals A, B we have sup(A + B) ≤ sup A + sup B?
2. [20 points]
(a) Exercise 2.1.3 from the textbook BA. Please argue using directly the definition of the
limit.
(b) Exercise 2.1.4 from the textbook BA. Please argue using directly the definition of the
limit.
(c) Exercise 2.1.5 from the textbook BA. Please argue using directly the definition of the
limit.
3. [20 points]
(a) Exercise 2.2.4 from the textbook BA.
(b) Exercise 2.2.7 from the textbook BA.
4. [20 points] Exercise 2.1.7 from the textbook BA. Please argue using directly the definition
of the limit.
5. [20 points] Exercise 2.2.9 from the textbook BA.
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