- The first midterm is on Friday, Apr 25, 12:00 pm, at MS 5148. The second midterm is on Friday May 23, 12:00 pm, at MS 5148.
- There will be five questions, all worth 10 points; only your best 4 out of 5 will be graded, so read the questions beforehand to decide which 4 to attempt (you may find that they are not all of equal difficulty). They will all be short proof type questions (similar to the homework questions, but somewhat shorter).
- The first midterm covers Weeks 1-3, up to uniform convergence but not the Weierstrass approximation theorem. The second midterm covers Weeks 4-7, covering differentiation but not any measure theory..
- The exam is open book, open notes; you may bring any course material or your own written notes. Bringing the solutions to your own homework may be a particularly good idea. Calculators are permitted but are very unlikely to be useful. No computers or other sophisticated electronic equipment is allowed.
- In addition to the open book and open notes, a page of definitions will be supplied with the midterm. This is so that you do not have to cycle through your notes just to find a particular definition.
- I expect your proofs to be fairly detailed - it should be clear to the grader which of the equations you write down are your assumptions, which ones are the things you are trying to prove, which ones are hypotheses, etc. Using English words such as "since", "assume", "by hypothesis", "by a theorem in the notes", etc. is highly recommended. On the other hand, you do not need to supply explanations for each and every step; for instance, the manipulations of high school algebra can be done without much comment. You do not need to give precise references (i.e. page numbers or Proposition numbers) for the results from the notes; it will be acceptable to say things like "By a theorem in the notes, ..." without specifying exactly which theorem. (This is, of course, assuming that the theorem you are quoting IS indeed in the notes).
- Unless otherwise specified in the question, you are allowed to use any result from the course notes or from the textbook when answering a question. You may also use any material from other classes (notably Math 131AH, but also Math 132, Math 121, etc.).

- There is no practice midterm; however I would recommend going through the homework assignments carefully prior to taking the midterm. In particular you should try to do as many of the homework problems as possible by yourself, without external aids (except for things like the class notes). You are encouraged to bring your own homework solutions to the midterm. (Bringing someone else's solutions (eg. the TA's) is much less likely to be helpful, unless you understand how the solutions work, in which case you probably didn't need to bring the solutions in the first place).
- Familiarity with the course notes is highly recommended; many of the proofs in the midterm may be easier if you use some of the propositions and theorems from the notes. Note that you should take some care reading the hypotheses of each proposition before using them; a proposition may apply to integers but not to rational numbers, or to convergent sequences but not to divergent sequences, or to finite sets but not to infinite sets, etc. This "fine print" is sometimes very important, so don't just apply the conclusion of the proposition blindly! You may find it helpful to create summaries of the course notes for quicker reference when taking the midterm; also the very process of creating summaries may help you organize the material mentally.
- Each midterm question is likely to bear some resemblance to a homework question, or perhaps a lemma in the notes, which suggests that the proof method is likely to be similar. For instance, if the midterm question looks like a homework question, and you used induction to do the homework question, it is likely that induction will be useful for the midterm question as well. However, while the methods may be similar, the actual content of the proof may be somewhat different, so merely copying a proof from the homework or notes word for word into the midterm is unlikely to impress the graders.
- It may help to write down exactly what the hypotheses are, and (in a separate location on the paper) write down what conclusions you want to reach. This may help suggest what propositions or proof methods may be helpful, and can help convince the grader that you know what you are doing. Note that one should always take care to separate the hypotheses from the conclusions; mixing the two up is almost always a bad idea.
- It may be helpful to use the definitions of terms to break up a complex statement into more elementary components. For instance, if one of the hypotheses is that "a_1, a_2, .... is a Cauchy sequence", it may help to use the definition of a Cauchy sequence to break this statement up into more elementary statements (e.g. using epsilon-steadiness). Of course, this can be taken too far... in most cases you don't want to go all the way back to natural numbers and ++, as most statements become ridiculously lengthy when expanded that way.
- The most common method to try to prove something is the direct or forward approach: write down your
hypotheses, see what you can derive from them, and eventually work your
way toward the conclusion. Sometimes, however, it is better to try
a backward approach: look at
the conclusion and think about
the possible ways one could get to that conclusion. This allws
you to replace your conclusion with successively easier conclusions,
working your way back towards something which will follow easily from
the hypotheses. A typical use of the backwards approach is to
take the conclusion, and use some definitions to break up any complex
concepts in that conclusion to simpler concepts. Often this makes
the conclusion more elementary, and thus easier to reach.

- In many cases a statement may look obviously true, and yet you do not know how to prove it. In this case you might ask yourself "is there any possibility that the statement is in fact false?". Often times you will find a good reason for why the statement cannot be false - and this means that you should probably prove your statement via contradiction.
- At certain times in a proof, you may want to use some result
which you are sure is true, but is not explicitly stated in the lecture
notes. In that case, you may wish to set that result up as a Lemma
(also called a Claim or an Auxiliary result), use that lemma to finish
the proof, and then go back and prove the lemma. Even if you don't
manage to prove the lemma, you can still get partial credit if you can
prove the original question assuming the lemma is true. One word of
caution, however - sometimes the lemma you wish to use is more
complicated than the original question, so this technique may not
necessarily save time. You should use your own judgement to see
if your lemma really is a more basic and fundamental fact than what you
are trying to prove.

- If you are completely stuck in proving a statement, you can at
least score partial credit by demonstrating to the grader that you at
least comprehend the statement.
Some ways in which you can do this is by drawing a picture, or
giving an example, or by explaining in informal English why you believe
the statement to be true, even if you can't explain it precisely enough
to qualify as a proof. Also, the very process of drawing a picture
or finding an example may lead you to understand better why the statement is true, and
perhaps give you an idea as to how to prove it.

- If you find that a proof is getting very complicated or very frustrating, you should set it aside and work on a different problem instead. The questions are intended to have relatively short proofs (unlike the homework, which is intended to be much more time-intensive), if you have the right approach and know what you are doing; so if you find yourself getting nowhere then you are probably on the wrong track. In which case it is often profitable to think about another question for a while and come back to it later.
- In this course it is important to know the
**precise**definitions of terms; if you misquote a definition when using it in a proof, you are likely to not be able to complete the proof correctly. Many of the questions in the Java Quiz will test your knowledge of various definitions; it might be worth giving that quiz a try just to identify your strengths and weaknesses.