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Study I

Write a study guide for yourself in preparation for Midterm I.
This is an optional assignment - if you are trying for the “Hold Me Accountable” grading scale (where I weight your exams a bit less in total points, by giving you points for studying/reflecting on them) then you should do this, and turn this in. If you do not want to write a study guide that is fine, it will not hurt you if you are going for the “I got this” grading scheme, where I’ll instead weight your exams higher and not ask for these smaller assignments along the way.

You can write the guide however helps you most, all that I ask is that your study guide shows serious effort of your studying for the exam, and you’ll get full points. Here are some possible suggestions if you are unsure what helps you best:

• Write down the important definitions for each topic you want to review. Explain them in your own words!
• For every definition, give yourself an example and a non-example
• Make a list of what theorems and propositions are related to which definitions.
• Come up with an example problem for each topic you want to review, and write out its full solution by hand! You’ll be handwriting solutions on the exam, so this is a good way to get yourself in that mindset and away from online work.
• For each topic you want to review, write to yourself how confident you are with it (maybe on like a 1-10 scale, if that’s helpful to you): this way when you’re studying you can look back and easily remember which things you were struggling with.
• If there’s multiple ways to do something, make a note of that for yourself! Try to find several different ways to do a single problem, and explain the differences in the solutions.

If you are working towards the “Hold me Accountable” grading scale, I will give you points for your study guide, and consequently the exam will be worth slightly less over all. If you are aiming to do this, note that your study guide should be clean, well written, and show clearly that you put a lot of effort into studying. Turning in a collection of scratch work for practice problems will not count for points here. (Though remember you do not need to write a study guide for points this is an optional assignment, and only matters if you are doing the Hold Me Accountable grading scale.)

List of Exam Topics

The exam will cover the material in the course so far (or will cover next week), which is all the book chapters through Monotone Convergence. Any section of a chapter marked with a star $★$ in my course notes is something we did not cover in detail in class, and I just wrote it for extra information for those interested: you’re not responsible for any of that. A (brief, probably incomplete) list of topics is below:

Numbers

• The field axioms, and their consequences
• The order axioms, and their consequences
• The completeness axiom
• Working with suprema and infima
• The archimedean property, infinitesimals, infinite numbers, and density of the rationals
• The Nested Interval Property, and uncountability of $\mathbb{R}$
• Existence of $\sqrt{2}$ in $\mathbb{R}$, and its proof (using suprema and the Archimedean property).

Sequences

• Definitions of Convergence, Divergence
• Proving a sequence converges directly from the definition
• Limits and inequalities: if $L\le a_n\le U$ then $L\le lim{a}_n\le U$
• The squeeze theorem
• The limit laws
  • Limits and constant multiples
  • Limits and sums / differences
  • Limits and products / quotients
  • Limits and square roots
• Applying these laws and theorems to calculate limits
• The Monotone Convergence Theorem: its proof, and using it to calculate limits.