Section B.3 Problem Sheet 3
Due 18 October at 3pm, either in class or in the pigeon hole in 4W.
Exercise 1.2.1 is there as a warm up, and is used in the proof of Lemma 1.21.
Exercise 1.2.2 gets you working with the definitions of open and closed sets in Definition 1.18 to prove Lemma 1.28, which will be useful in several places. The last part of the exercise asks you apply this lemma to a very concrete problem; the answers are perhaps different than what you would initially expect! Even if you have trouble with the first or second part, I would encourage you to see if you can do this last part.
Exercise 1.3.1 collects some useful rules for computing limits, something like the so-called ‘algebra of limits’ you see in Analysis 1. Figuring out how to prove them should hopefully give some additional practice working with limits and inequalities.
Finally, Exercise 1.3.2 is a companion to Exercise 1.1.9 on the previous sheet. It gives you a bit of practice thinking about convergence in a slightly new setting, and we’ll also use it several times throughout the unit. Like Exercise 1.2.1 it is quite short, and so this week I’m assigning five exercises instead of four.
There are lots of other good exercises to try in Section 1.1, Section 1.2 and Section 1.3, and as always I would encourage you to attempt additional exercises on any topics that you want more practice with.
Please feel free to email me, or drop by office hours (Tuesdays 1:15–2:05 in 4W 1.12), with any questions about these problems whatsoever. I am also more than happy to meet one-on-one or in a small group.
