Online Examination–V1–Fundamentals of Real Analysis
Course Coordinator – Regina S. Burachik
SP2 2021 MATH 2027
1. Justify every assertion with a proof or a valid quote, and show your work and rationale in all questions. I like to give partial credit, so try to make it easy for me to give it to you, by clearly explaining what you are doing. In your responses, feel free to quote any result from the book to support your claims. A valid quote used to justify a claim is as good as its proof, and will grant you full marks for that particular claim. So use facts from the book as much as you can.
2. Notation is the same as that used in the textbook.
3. This exam has 4 pages.
4. Total Marks: 100
Recall (see Definition 2.2.4) that the ε-neighbourhood of a ∈ R is the set Vε(a) = {x ∈ R : |x−a| < ε}, and that (see Definition 3.2.11) the closure of a set A ⊂ R is defined as A = A ∪ L(A), where L(A) := {x ∈ R : ∀ ε > 0, A ∩ Vε(x) contains points different than x} is the set of limit points of A (see Definition 3.2.4). Recall (see Exercise 3.2.14) that A◦ := {x ∈ A : ∃ ε > 0 such that Vε(x) ⊂ A} is the interior of A. Denote by iso(A) the isolated points of the set A, defined as those points a ∈ A for which thereexistsε>0suchthatVε(a)∩A={a}.GivenasetS⊂R,recallthatSc :={x∈R : x̸∈S}.
Question 1. (Topology of R, open and closed sets, closure of a set)
(a) [3POINTS]Showthata∈Aifandonlyifforeveryε>0wehaveA∩Vε(a)̸=∅.
(b) [3 POINTS] Let I be an arbitrary nonempty set. Show that
[Ai ⊂[Ai. (1)
i∈I i∈I Hint: use the characterization given in part (a).
(c) [2 POINTS] Given n ∈ N, define An := √2,√2!. Show that Sn∈N An = (0,√2). 3n
(d) [2 POINTS] Use the family of sets in part (c) to show that the opposite inclusion in (1) is false.
(e) [1+2=3 POINTS] Consider now a single point x ∈ R. Show that a set Sx := {x} consisting of a
single point is a closed set (hint: check its complement). Use the family given by Sn := (√2) 3n
to contradict again the opposite inclusion to (1). In other words, show that
[ Sn ( [ Sn. n∈N n∈N
Hint: Note that Sn∈N Sn = { √2 : n ∈ N} and compute the closure of this union (check Example
3.2.9(i)). [3+3+2+2+3=13 POINTS]
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Online Exam–V1–Fundamentals of Real Analysis MATH 2027 2 Question 2. (Supremum and Infimum, Sequences and series in R, Topology of R)
Lety1 =3,andforeachn∈Ndefineyn+1 :=(2yn −3)/3.
(a) [3 POINTS] Use induction to prove that the sequence (yn) satisfies yn > −3 for all n ∈ N.
(b) [3 POINTS] Use induction to prove the sequence (yn) is strictly decreasing.
(c) [2 POINTS] Explain why (a) and (b) imply that the sequence (yn) converges.
(d) [2 POINTS] Compute lim yn. n→∞
(e) [5 POINTS] Let T := {yn : n ∈ N}. Compute T, T◦,L(T) and iso(T). Is the set T compact? Justify your answers.
(f) [2 POINTS] Find sup(T ) and inf(T ). Is sup(T ) = max(T )? Is inf(T ) = min(T )? Justify your answers.
(g) [3 POINTS] Show by induction that the sequence (yn) defined above verifies that 2n
yn =−3+3n−2, foralln∈N.
(h) [4 POINTS] Define an := (yn + 3) . Use part (g) to show that the two series given by P an and
P(−1)nan converge. 9 P
(i) [2 POINTS] Show that the series yn diverges.
[3+3+2+2+5+2+3+4+2=26 POINTS]
Question 3. (Supremum and Infimum)
Decide if the following statements about suprema and infima are true or false. In all cases, the sets A and B are nonempty subsets of the real line R. Give a short proof for those statements that are true. For any that are false, supply an example where the statement does not hold.
(a) [3POINTS]IfA(B,andBisboundedbelow,theninfB>infA.
(b) [3POINTS]IfsupB