UNSW SCHOOL OF MATHEMATICS AND STATISTICS
MATH3570 Foundations of Calculus
Term 1, 2024 Assignment, version B
Due 10th March 2024
Instructions
See the Assignment Brief or list of FAQ on moodle for further details. Three brief re-
minders:
ˆ This assignment is worth 20% of your final mark. Late submission is not permitted
without an approved special consideration submission.
ˆ Your solutions to this assignment need to be properly typed. LATEX is preferred, but
Word or something similar is acceptible. You should then submit your solution as
a PDF file (or files) via Moodle. Sample files in a suitable format are available on
Moodle.
ˆ Submissions may be computer checked for signs of copying.
Questions
1 [5 marks] Consider the statement
∀x > 0 ∃y > x ∀z ∈ (y, y + 1), 0 < sin z − sin y < z − y .
a. Write the negation of the above statement.
b. Which is true, the statement or its negation? Prove your answer.
2 [5 marks] Prove from the definition of limit that
lim
n→∞
3n+ 5 sinn
7n− sinn =
3
7
.
3 [7 marks] Consider the sequence an+1 =
46(an + 44)
an + 46
and suppose that a1 = 48.
a. Prove by induction that an >
√
2024 for n ≥ 1.
b. Prove that {an} is a monotonic decreasing sequence and so converges. Find its
limit, explaining why your answer is correct.
Questions continue overleaf
1
4 [5 marks] Let A and B be two non-empty subsets of R with A ∩B not empty.
a. Show that if A and B are bounded below then
inf(A ∩B) ≥ max {inf(A), inf(B)} .
b. Give an example where the inequality is strict and explain why your example
works.
Marks for presentation
Marks for quality of mathematical presentation will be awarded based on the following
rubric:
3 marks If there are no or very few typographical and grammatical errors.
2 marks If there are several typographical and grammatical errors or some unclear explana-
tions.
1 marks If there are major presentation issues or several questions had unclear explanations.
0 marks If there were no sentences or explanations provided in the question responses.
Total Marks
The maximum mark for this assignment is 25.