MA10207 Analysis 1A (Winter ’11)
Tutorials
I give tutorials at 14:15 on Fridays in 3W 3.25b, for groups A1 and C6. Work should be handed in to my folders in the pigeonholes for the module on the first floor of 4W, at 12:15 on Thursdays (except for problem sheet 0, which should be handed in at the same time on Tuesday of week 2).
Moodle
The Moodle page for this module can be accessed here. Go there to access problem sheets, notes and additional reading for the course.
Week 1
For a discussion of the truth table for the implication sign, click here.
Week 2
For an example of a proof that a sequence converges to a given limit, click here.
Week 3
Not much to say this week as we just went over the second problem sheet. If you don’t remember/understand what I said about it, then check the model solutions on Moodle.
Week 4
This week’s sheet is mostly about completeness, and related ideas. I’ve written some general comments about this, with the problem sheet questions in mind.
Week 5
This week’s comment explains how you can come up with clever tricks in a proof, by going into the example from today in more detail. I’ve also included a warning about what can happen when you apply theorems backwards.
Week 6
Most of you were happy using the ratio and comparison tests for series, although you must remember that the \(\sum\)s are vital to the notation—\((\frac{1}{n})\) is not a divergent sequence, but \(\sum_{n=1}^\infty\frac{1}{n}\) is a divergent series. I’ve written something about how you might find the right constants in the solution to Q2.
Week 7
When doing this week’s problem sheet remember that you can only use induction to show something for all natural numbers, not the infinite case. So you can use induction to prove the ‘generalised triangle inequality’ in Q1, but not the ‘infinite triangle inequality’ in Q2. Your convergence tests will be useful in the other questions—make sure you check whether the series converges or diverges for all \(x\).
Week 8
There was some confusion about the proof of Cauchy’s multiplication theorem, so I’ve written out another version, trying to put particular emphasis on the parts I expect are most confusing.
Week 9
The definition you have of a function being sequentially continuous with respect to \(\mathbb{R}\) is slightly confusing, and you have to unpack it. Note that you say a function \(f\) is sequentially continuous with respect to \(\mathbb{R}\) at a point \(x\) if for all real-valued sequences \((x_n)\) converging to \(x\), the sequence \((f(x_n))\) converges to \(f(x)\), and simply sequentially continuous with respect to \(\mathbb{R}\) if this is true for all points \(x\in\mathbb{R}\). So if you want to show a function is sequentially continuous with respect to \(\mathbb{R}\), then you should show that for any point \(x\), and any real-valued sequence \((x_n)\) converging to \(x\), the sequence \((f(x_n))\) converges to \(f(x)\). Note that \(x\) is now arbitrary, unlike in the definition of sequential continuity at the point \(x\). A good exercise to help you understand these ideas is to prove that the Heaviside step function is not sequentially continuous with respect to \(\mathbb{R}\).
Week 10
If you haven’t already done so, now would be a good time to start looking at the past papers for the module on Moodle, particularly the 2010/11 paper. Make sure you can state all the definitions and key theorems, as these will be easy marks (and are important things that it’s a good idea to know anyway).
Revision Week
There will be a drop-in session for this module on Friday 13th January at 10:15 in 8W 2.1. Please bring along any issues you have with the material covered in lectures, or any specific problems from the examples sheets and past exam papers that you had trouble with. Good luck with the exam!