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MAST20026 Assignment
This assignment consists of 2 pages.
Due Date: Friday May 13
Assignments are due in Gradescope on the due date listed above. To ease submission, please prepare assignment
solutions for each question on its own page. Do not include your name anywhere on your assignment.
Students are encouraged to work together on understanding the problems and their solutions. However, submitted
solutions must be prepared individually, in your own words, and without the aid of others. Answers presented
without justification will receive no marks.
If you do not completely understand your submitted solutions, it is possible you are committing academic
misconduct. This is a serious offense.
Please do not upload this assignment paper or any of its questions to any online “help” service (e.g., CourseHero,
Chegg, etc...). By doing so, you are actively ruining the learning experience for yourself and future students and
more than likely violating the University’s academic misconduct policy.
PART A
This semester a PhD researcher is conducting a study into the effectiveness of having students peer-review
submitted work.
On Assignment 3 you were asked to prove a version of the Completeness Axiom for infima. On this assignment you
are tasked with reviewing anonymous solutions to this question from students in a previous cohort. Submit your revised
solution together with a short reflection (approx 100 words) that comments on how your revised solution differs
from your initial solution. Your reflection should answer the following, as appropriate:
• What changes did you make between your first and second solution and why?
• What did you decide to keep the same between your first and second solution and why?
• What new knowledge did you take away from the peer review activity? Did it help you understand
the problem better? Did it help you understand how to communicate your thinking more clearly to the
reader?
If you did not make a submission for Assignment 3, prepare a solution to Question (4) from Assignment 3 and
comment on how doing the peer-review activity helped to create your solution.
Though this activity is designed to as a means for the PhD researcher to collect data, there is no obligation
for you to have your data included as part of the research project. Once you have done the peer-review work
(by visiting the link above) you will be asked if you agree to include your data as part of the research project.
Whether you agree or disagree to take part in the study will have no impact on your assignment grade.
5 marks
PART B
(2) (4 marks) Let f : R → R be a function. Consider the sequence (fn) so that fn = f(n) for all n ∈ N+.
Let L ∈ R.
(a) Prove that if lim
x→∞ f(x) = L, then limn→∞ fn = L.
(b) By giving a counter example, prove the converse of the implication in the previous part is false, You
do not need to prove your counter-example is correct. However you should write a sentence or two
that convinces the grader than your counter example is indeed a counter example to the statement
of the converse.
(3) (5 marks) Let E ⊆ R, let f : E → R, let a be limit point of E and let L ∈ R. Using an -δ argument,
prove that if lim
x→a−
f(x) = lim
x→a+
f(x) = L, then lim
x→a f(x) = L.
This proof is part of the proof of Theorem 5.7. No marks will be awarded for solutions that invoke Theorem
5.7.
(4) In this exercise we prove the following theorem
1
Theorem. Let E ⊆ R and let a ∈ R. If a is a limit point E, then there exists a sequence f : N+ → E\{a}
such that fn → a.
(a) (1 mark) Briefly explain how you know that for each δ > 0 there is at least one solution for
0 < |x− a| < δ among the elements of E.
(b) (5 marks) For fixed δ > 0, let xδ ∈ E denote a solution for 0 < |x − a| < δ. For example, x = x0.8
is a solution for 0 < |x− a| < 0.8. Consider the sequence
(fn) =
(
x1, x 1
2
, x 1
3
, x 1
4
, . . .
)
Using an -M argument, prove this sequence converges to a.
(5) (5 marks) Prove the converse of the theorem in the previous question.
PART C (Optional – nothing to submit, no marks, you will not be assessed on the content of PART C)
• The LATEX source file for this assignment can be found in the Assignment 4 folder in Files on Canvas.
This is a great starting point if you want to prepare your solution for this assignment using LATEX.
• In this folder you will also find a named Assignment4PartC.tex. Open and compile this file in your LATEX
editor and read about tables and alignment.
• Find the input that gives the following outputs.