Math 201-B
Introduction to Proofs
Instructor: Alex Roitershtein
Iowa State University
Department of Mathematics
Fall 2015
Exam #3 (practice test)
December 18, 2015
Student name:
-
Student ID:
Duration of the exam 120 minutes.
The exam includes 5 questions.
The total mark is 100 points.
Please show all the work, not only the answers.
Calculators, textbooks, and help sheets are allowed.
1. [20 points]
(a) Let A and B be bounded subsets of R such that A ⊂ B. Show formally that sup A ≤ sup B
and inf B ≤ inf A.
(b) Let
A = {x ∈ R : x = n−s for some n ∈ N, n ≥ 2 and s ∈ R, s > 0}.
Show that sup A = 1 and inf A = 0.
2. [20 points]
(a) Exercise 2.1.6 from the textbook BA.
(b) Exercise 2.1.13 from the textbook BA.
(c) Exercise 2.1.15 from the textbook BA.
3. [20 points]
(a) Exercise 2.2.3 from the textbook BA.
(b) Exercise 2.2.8 from the textbook BA.
4. [20 points] Exercise 2.1.8 from the textbook BA. Please argue using directly the definition
of the limit.
5. [20 points] Exercise 2.2.6 from the textbook BA.
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