AS/2241/1/2022-2023
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2241 Introduction to Mathematical Analysis
Assignment 1
Due date: Sept 23 (Fri), 2022 before 5:00 p.m.
Instructions
• No late submission will be accepted.
• Please scan your work properly in A4 size and submit it as a PDF file
in moodle. Oversized, blurred or upside-down images will NOT be
graded.
Qn. 1 (8 points) Prove that for any a, b ∈ R, if a < b + ϵ for any ϵ > 0, then
a ≤ b.
Qn. 2 (12 points)
(i) Suppose that A and B are subsets of R which are nonempty and
bounded below. Prove that inf(A ∪ B) = min{inf A, inf B}.
(ii) Suppose that A and B are subsets of R which are nonempty and
bounded above. Prove that sup(A ∪ B) = max{sup A, sup B}.
Qn. 3 (25 points) Prove or disprove each of the following statements:
(i) Let A be a nonempty subset of R. If A is bounded above, then the
complement Ac ( =R \ A) of A is bounded below.
(ii) Let A and B be nonempty bounded subsets of R. If a ≤ b for any
a ∈ A and any b ∈ B, then sup A ≤ inf B.
1
(iii) Let A and B be nonempty bounded subsets of R. If a < b for any
a ∈ A and any b ∈ B, then sup A < inf B.
(iv) There exists a nonempty subset A of Q such that sup A = 4 and
4∈
/ A.
(v) The set A = {a ∈ R : a = πr for some r ∈ Q} is dense in R.
Qn. 4 (15 points) By adapting the proof of the Archimedean Property, prove
that the set {2n : n ∈ N} is not bounded above. (Do not make any use
of logarithm.)
Qn. 5 (10 points) Prove that sup{r ∈ Q : r < a} = a for any a ∈ R.
Qn. 6 (15 points) For each n ∈ N, let In = [an , bn ] be a nested sequence of
closed and bounded intervals. Prove that
∞
\
In = [a, b]
n=1
where a = sup{an : n ∈ N} and b = inf{bn : n ∈ N}.
Qn. 7 (15 points) For each n ∈ N, let P (n) be a statement about the natural
number n. Suppose that
(i) P (1) is true; and
(ii) for any k ∈ N, if P (k) is true, then P (k + 1) is true.
Prove that P (n) is true for all n ∈ N.
Hint. Let A = {n ∈ N : P (n) is false} ⊆ N . Show that it’s empty by
considering its infimum.
End of Assignment 1
2