MAST20026 Real Analysis
Semester 1 Assessment, 2025
Question 1 (9 marks)
Let p and q be primitive statements.
(a) Write a truth table for the compound statement
q ⇒ (p ∨ (∼ q)).
(b) If q is true and q ⇒ (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) Find a statement with exactly one connective ∼, ∧, ∨, ⇒ , or ⇐⇒ which is logically equivalent to q ⇒ (p ∨ (∼ q)). Justify your answer using a truth table.
Question 2 (9 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) together with the fact that 0x = 0 for all x ∈ R, write a (semi-)formal proof showing that for all a, b ∈ R
if a > 0 and b < 0, then ab < 0.
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 3 (9 marks)
Let A and B be subsets of R which are both non-empty and bounded above. Define
A + B = {x + y : x ∈ A and y ∈ B}.
(a) Show that if α is an upper bound of A and β is an upper bound of B, then α + β is an upper bound of A + B.
(b) Show that sup(A + B) = sup A + sup B.
Question 4 (12 marks)
For each of the following situations, provide a specific example for which the situation is true, and briefly justify your answer.
(a) A set E ⊆ Q such that the set of all limit points of E is [0, ∞).
(b) A sequence (fn) such that () converges to 9 but (fn) diverges.
(Here = fn · fn for each n ∈ N+.)
(c) A convergent series such that diverges.
Question 5 (13 marks)
Let (xn) be the sequence such that x1 = 0 and
for each n ∈ N+.
(a) Using a proof by induction, show that 0 ≤ xn < 3 for all n ∈ N
+.
(b) Is the sequence (xn) monotone increasing, monotone decreasing, or neither? Justify your answer.
(Hint: Compute xn+1 − xn.)
(c) Is the sequence (xn) Cauchy? Briefly justify your answer.
Question 6 (11 marks)
Let f : R → R be the function defined by
(a) Using the − M definition of limits, show that f(x) = 1/3.
(b) Using the − δ definition of continuity, show that f is not continuous at 2.
Question 7 (13 marks)
Let f : [−1, 1] → R be the function defined by
(a) Let P = {x0, x1, x2, . . . , xn} be any partition of [−1, 1]. Compute the lower Riemann sum L(f, P).
(b) Find a positive lower bound for the upper Riemann integral of f on [−1, 1].
(c) Is f Riemann integrable on [−1, 1]? Justify your answer using Parts (a) and (b).
Note. For Question 8, you may assume without − M proofs that limits behave the way you expect.
Question 8 (14 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