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 Deinition 2.2.4) that the ε-neighbourhood of a ∈ R is the set Ve (a) = {x ∈ R : |x−a| < ε}, and that (see Deinition 3.2. 11) the closure of a set A ⊂ R is deined as A = A ∪ L(A), where L(A) := {x  ∈ R   :  ∀ε  > 0,  A ∩ Ve(x) contains points different than x} is the set of limit points of A (see

Deinition 3.2.4). Recall (see Exercise 3.2. 14) that A◦  := {x ∈ A : ∃ ε > 0 such that Ve(x) ⊂ A} is the

interior of A. Denote by iso(A) the isolated points of the set A, deined as those points a ∈ A for which

there exists ε > 0 such that Ve(a) ∩ A = {a}. Given a set S ⊂ R, recall that Sc  := {x ∈ R  : x \∈ S}.

Question 1. (Topology of R, open and closed sets, closure of a set)

(a)  [3 POINTS] Show that a ∈ A if and only if for every ε > 0 we have A ∩ Ve(a) ∅.

(b)  [3 POINTS] Let I be an arbitrary nonempty set. Show that

n Ai ⊂ n Ai .                                                        (1)

i∈I             i∈I

Hint: use the characterization given in part (a).

(c)  [2 POINTS] Given n ∈ N, deine An  := ( , √2). Show that nn∈N An = (0, √2).

(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  := { }

to contradict again the opposite inclusion to (1). In other words, show that

n Sn 车 n Sn . n∈N             n∈N

Hint: Note that nn∈N Sn = { : n ∈ N} and compute the closure of this union (check Example

3.2.9(i)).

[3+3+2+2+3=13 POINTS]

Question 2. (Supremum and Inimum, Sequences and series in R, Topology of R)

Let y1 = 3, and for each n ∈ N deine yn+1  := (2 yn − 3)/3.

(a)  [3 POINTS] Use induction to prove that the sequence (yn ) satisies 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→8

(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 ) deined above veriies that

yn = −3 + ,  for all n ∈ N.

(h)  [4 POINTS] Deine an  :=       9      . Use part (g) to show that the two series given by对 an and

对(− 1)n an converge.

(i)  [2 POINTS] Show that the series 对 yn diverges.

[3+3+2+2+5+2+3+4+2=26 POINTS]

Question 3. (Supremum and Inimum)

Decide if the following statements about suprema and inima 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)  [3 POINTS] If A 车 B, and B is bounded below, then inf B > inf A.

(b)  [3 POINTS] If sup B < inf A, then A ∩ B = ∅.

(c)  [6 POINTS] Assume that B ⊂ A and that A is bounded above and verifying the following property: A x ∈ A there exists y ∈ B such that y ≥ x.

In this situation, we must have sup(B) = sup(A).

[3+3+6=12 POINTS]

Question 4. (Cardinality, countable and uncountable sets)

Denote by H the set of positive numbers which are multiples of 11, namely H  := {t ∈ N  :  3 k ∈ N such that t = 11k}. Use results from the book or a direct proof to explain why the following claims hold.

(a)  [4 POINTS] There is no injective function f : ( − 1, 1) → H.

(b)  [4 POINTS] There is a bijective function f : H → Q.

[4+4=8 POINTS]

Question 5. (Supremum and Inimum, Topology of R, Continuity of functions)

Give an example of the situation described or state that there is no such example. When there is no such example, provide a compelling argument for why this is the case. In all cases, the sets are subsets of the real line R.

(a)  [4 POINTS] Two nonempty sets A and B with A ∩ B  =  ∅, inf A  =  inf B , inf A  \∈ A, and inf B \∈ B .

(b)  [4 POINTS] Two functions f and g, neither of which is continuous at x = 2, but f(x)(g(x))5 and f(x) + 2g(x) are continuous at x = 2.

(c)  [2 POINTS] A function f discontinuous at x = 2, but sin (f(x) + T) is continuous at x = 2.

(d)  [6 POINTS] Two nonempty sets C and D, with C bounded above, C = D, and sup C < sup D. [4+4+2+6=16 POINTS]

Question 6. (Sequences, Cauchy sequences)

Let (xn ) and (yn ) be two real sequences. In parts (a)–(c), assume that (xn ) is a Cauchy sequence and that (yn ) is bounded. Decide whether the following sequences are or not Cauchy. To justify your answers, you may quote a result from the book or provide a direct proof. Provide a counterexample if you decide the sequence is not Cauchy.

(a)  [4 POINTS] The sequence (an ) deined by an  := (3xn + ( − 1)  )n.

(b)  [4 POINTS] The sequence (bn ) deined by bn  := ((xn )2 yn ).

[4+4+4=12 POINTS]