Second Semester, 2021 Mathematical Sciences Institute
Australian National University
MATH1116, Advanced Mathematics and Applications 2
Assignment 2
Most of this Assignment is WebAssign as per previous notification.
Your WebAssign questions for Assignment 2 are due by Sunday 8th August.
You do not need to provide any working for the WebAssign questions.
Your submissions for the single ‘Show Working’ question below should also be
submitted to Gradescope by Sunday 8th August — it should be fairly quick
to complete this one written out task, but email Griff if you missed it (or
don’t have time) because this single question was released late.
• Important: please respond to the Academic Integrity Agreement that is linked to
from the Week 2 block on Wattle. You will need to select “I agree” to be able to be
given any marks for assessment in MATH1116 this semester.
• If you have not already done so, you should carefully read the information in the first
several pages of the Assignment 1 PDF, available in the Week 1 block on Wattle.
WebAssign
See the WebAssign platform for the questions — worth 44 marks out of 50 for this
Assignment. Read the ‘Description’ and the ‘Instructions’ at the top of the quiz for
further details.
To login to WebAssign you should go to https://www.webassign.net/wa-auth/login
and use your university email address as your username, with the password that you have
previously set (not your main ANU password).
If you need to set up your password or have forgotten it, you can use the password reset
process from the login page.
Analysis
The single ‘Show Working’ question (next page) asks you to generalise a theorem and proof
(taken from the Limits and Continuity Theory Notes on Wattle) from single variable R to
multivariable Rn. You need to come up with the generalisation of the theorem yourself,
and write out the proof with the appropriate modifications. You can mostly copy the
provided proof, you just need to figure out where changing an absolute value symbol to a
norm symbol is appropriate, and where absolute value symbols should be left as they are!
https://www.webassign.net/wa-auth/login
MATH1116, Assignment 2 2
Question 1. Generalisation to Rn (Show Working) 6 points
Generalise the following theorem and proof so it applies to x, y ∈ Rn in place of of x, y ∈ R.
Theorem. Suppose x, y ∈ R. Then
|x| − |y| ≤ |x− y| and furthermore
∣∣∣|x| − |y|∣∣∣ ≤ |x− y| .
Proof. By the triangle inequality, we have
|x| = |(x− y) + y| ≤ |x− y|+ |y| ,
hence
|x| − |y| ≤ |x− y| .
Similarly,
|y| = |(y − x) + x| ≤ |y − x|+ |x| ,
hence
|y| − |x| ≤ |y − x| = |x− y| .
Thus both |x| − |y| and |y| − |x| are less than or equal to |x− y|, which gives∣∣∣|x| − |y|∣∣∣ = max {|x| − |y| , |y| − |x|} ≤ |x− y| . �