MAST20026 Real Analysis
Semester 2 Assessment, 2024
Question 1 (8 marks)
Let p and q be primitive statements.
(a) Write a truth table for the compound statement
(∼ p) ⇐⇒ (p ∨ q).
(b) If (∼ p) ⇐⇒ (p ∨ q) is true, what can you conclude about the truth value of p? Justify your answer in terms of specific lines in truth table from Part (a).
(c) Using a truth table, show that (∼ p) ⇐⇒ (p ∨ q) is logically equivalent to ∼ (q ⇒ p).
Question 2 (9 marks)
Find the largest positive integer d such that for every n ∈ N, 5n − 8n − 1 is divisible by d. Use induction to prove your assertion is true.
(Hint: Start by writing out 5n − 8n − 1 for n = 0, 1, 2, 3 and making a reasonable guess for the value of d.)
Question 3 (7 marks)
Listed below are the Axioms of the Real Numbers.
O1. ∀x, y ∈ R exactly one of the following is true: x < y, y < x, or x = y.
O2. ∀x, y, z ∈ R if x < y and y < z then x < z.
A1. ∀x, y ∈ R x + y ∈ R
A2. ∀x, y ∈ R x + y = y + x
A3. ∀x, y, z ∈ R (x + y) + z = x + (y + z)
A4. ∃0 ∈ R ∀x ∈ R x + 0 = x
A5. ∀x ∈ R ∃ − x ∈ R x + (−x) = 0
M1. ∀x, y ∈ R xy ∈ R
M2. ∀x, y ∈ R xy = yx
M3. ∀x, y, z ∈ R (xy)z = x(yz)
M4. ∃1 ∈ R \ {0} ∀x ∈ R 1x = x
M5. ∀x ∈ R \ {0} ∃ 1/x ∈ R \ {0} x · 1/x = 1
D. ∀x, y, z ∈ R x(y + z) = xy + xz
OA. ∀x, y, z ∈ R if x < y then x + z < y + z
OM. ∀x, y ∈ R if x > 0 and y > 0 then xy > 0
C. Every non-empty subset of R that is bounded above has a least upper bound.
Using only the Axioms of the Real Numbers (and rules of logic), write a formal proof showing that for all non-zero a ∈ R
1/(1/a) = a.
Note. Here and for the remainder of the exam, you may write informal proofs, and you may assume that algebra and inequalities with the real numbers behave the way you expect.
Question 4 (9 marks)
For each of the following situations, provide a specific example for which the situation is true, and briefly justify your answer.
(a) An unbounded set E such that the set of all limit points of E is [0, 1].
(b) A non-constant sequence (fn) which is contractive and converges to −1.
(c) A function g : R → R such that |g| is continuous on R but g is not continuous at a for all a ∈ R.
Question 5 (12 marks)
Let A, B ⊂ R be sets which are both bounded above in R. Let α be the least upper bound of A and β be the least upper bound of B.
(a) Show that max{α, β} is an upper bound of the set A ∪ B.
(b) Show that sup(A ∪ B) = max{α, β}.
(c) Is it always true that max{α, β} is a limit point of A ∪ B? Justify your answer with a proof or counter-example.
Question 6 (10 marks)
Consider the sequence (fn) defined by
for each n ∈ N+.
(a) Show that (fn) is a monotone increasing sequence.
(b) Use the − M definition of convergence to show that
(c) Is the sequence (fn) Cauchy? Justify your answer using definitions and theorems from lecture.
Question 7 (11 marks)
Let f : R → R be the function defined by
(a) Using the − δ definition of continuity, show that f is not continuous at 0.
(b) Using an − δ argument, show that f is differentiable at x = 1.
Question 8 (12 marks)
Let f : [−1, 1] → R be defined by
Let n ∈ N+ and let P be the partition
(so that xi = −1 + i/n for each i).
(a) Draw a picture of illustrating how to compute the lower Riemann sum L(f, P) when n = 4. Include the graph of f, the partition P of [−1, 1], and the rectangles for each partition. The sum of the area of these rectangles should equal L(f, P).
(b) Let n ∈ N+. Compute the lower and upper Riemann sums with respect to the partition
(so that xi = −1 + i/n for each i).
(c) Is f Riemann integrable on [−1, 1]? Explain.
Note. For Questions 9 and 10, you may assume without − M proofs that limits behave the way you expect.
Question 9 (13 marks)
(a) For each of the following infinite series, use an appropriate test or theorem to prove that the series converges or diverges.
(b) Find the radius of convergence and interval of convergence for the power series
Question 10 (9 marks)
Consider the function f : (−8, ∞) → R defined by
for each x > −8.
(a) Find the Taylor series expansion of f at 0.
You do not need to justify your formula for the coefficients by a rigorous proof by induction.
(b) Show that the Taylor series of f converges to f on the interval [−4, 4].