Request an exercise

There are a lot of exercises in Analysis I (if I counted correctly, there are 313 exercises in the main text, plus 13 more in the appendixes), and it will take me a long time to get to all of them. You can use this page to request that a specific exercise be prioritized, so that I will work on it sooner than otherwise. Just add a comment to this page with the exercise number.

157 thoughts on “Request an exercise”

  8. Could you please verify that I can prove Exercise 3.5.13 as follows: define the set of all functions f:N\rightarrow N' (this set exists due to the power set axiom), and then apply the axiom of specification to “single out” the functions f in that set such that f(0)=0' and for all n\in N and n'\in N', f(n)=n'\leftarrow\rightarrow f(n+1)=n'+1' (i.e. pin down the functions f that obey such properties). Show that the resulting specified set is singleton by essentially showing that f is unique, so then one takes its union (which exists by the union axiom), and then one shows inductively that this unique $f$ is injective and surjective, which would make it a bijection and therefore resolve the exercise without invoking Exercise 3.5.12 as the hint suggests.

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  12. I would like to see a solution to Exercise 6.5.3. which uses the hint Tao gives. I asked the question on the Mathematics Stack Exchange earlier and got some good answers there, but none of them really seemed to invoke this hint, though I can sort of tell that Tao had plunged into the eventually $\epsilon$-close definition of convergence to a limit, etc.
    https://math.stackexchange.com/questions/3959574/if-x0-is-a-real-number-show-that-lim-n-rightarrow-inftyx1-n-1?noredirect=1#comment8165848_3959574

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    1. I’d like to finish blogging all of Analysis I before I even consider blogging exercises from Analysis II, so it will be a very long time before I get to this one.

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  24. I’m in high school and wanted an add-on that would help me find the ones in chapter 5 on my own, I’m having difficulty with these inequality tricks used in examples 5.2.1 and 5.2.2, among others.

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    1. Can you say more about what you thought of my post for Exercise 5.2.1? Was there something unclear? I thought I gave a pretty good stream of consciousness for how to come up with the proof there.

      Lara Alcock’s How to Think about Analysis or Kenneth A. Ross’s Elementary Analysis might have what you are looking for, but it’s been a while since I looked at those books (I just remember they did things slightly more slowly/gently than other books).

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  25. the problem was just the little inequalities trick, for example | b_j – b_k | = | b_j – a_j + a_j – a_k + a_k + b_k |

    This kind of trick I couldn’t identify on my own, as I haven’t worked much with inequalities yet.

    Your proof is very readable and understandable, don’t worry

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    1. Ok I see. I don’t think there is a book that will go through such tricks and show you how to find them, though that would be very interesting if it existed. Instead you will see these tricks used repeatedly by other people and then you will start to use them yourself.

      I also just remembered the Tricki, though the tricks there might be more advanced.

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  30. 3.3.6 my proof:

    Part 1:
    Let x ∈ X. Since f is bijective, there exists exactly one y ∈ Y such that:
    f(x) = y

    f^(-1)(y) := x by remark 3.3.25.

    => f^(-1)(f(x)) = x for all x ∈ X.

    Part 2:
    Let y ∈ Y. Since f is bijective, there exists exactly one x ∈ X such that:
    f^(-1)(y) = x. (?)

    => f(f^(-1)(y)) = y for all y ∈ Y.

    My question is: isn’t (?) that we’re require to proof?

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  32. Hello, Your solutions to the exercises are great and helped me a lot. Can You please solve exercises 6.4.10, 6.5.1 and 6.5.3 or just some of them. Thank You 🙂

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  34. Exercise 3.3.4 please. I also have a question about the proof of the third part. I use the definition of surjective, and can we just substitute f(x) with y directly. I’m not sure whether it is permitted.

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    1. I’d like to finish blogging the solutions to volume 1 first, so at my current rate it will be quite some time before I even get started on volume 2.

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  39. Please solve some excercises from chapter 8 as it is a very important topic(especially with the introduction of Axiom of Choice and the Zorn’s Lemma). My personal request is question 8.5.5 in which we have to find conditions under which pre image preserves total ordering.
    Thank you.

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    1. In principle I would love to accept contributions from others, but in practice it’s been quite challenging to make it work. I’ve had maybe 5 people contact me over the years about this, and in at least two cases got as far as reviewing solutions people have written, but in those cases it’s still involved quite a bit of back and forth to get the solutions to a state I am happy with, which meant that I was still spending a lot of time on the solutions (i.e. it didn’t lead to saving me much time in the end, which defeats the purpose of having people help me). So my default recommendation would be to just start your own blog and post your solutions there. Sorry about the discouraging message here, but I also don’t want to falsely get your hopes up too much, given my previous experience with this.

      If you still want to attempt to contribute to this blog in particular, then you are welcome to send me solutions at riceissa@gmail.com. Please keep in mind that as the blog is intended for students working through the book (who are often just learning how to write proofs for the very first time), I want all the posts to be written in prose like the ones in the book, mimic the section headings given in the template page, have a pedagogic tone with lots of intuition and tips on how to prove things, etc. (It’s always tricky to be accepting/inclusive while also maintaining a quality bar…) I hope that makes sense!

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    2. Dear Issa Rice. Thank you for replying, and your concern absolutely makes sense. inspired by several blogs including yours I feel it is better that I build a blog for myself. I will send you an email once I have some reasonable material in my blog. Thanks a lot!

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  49. Tao 1,chapter on sets,Tao says:
    For all x,x is not an element of A and x is not an element of the empty set.
    Later he goes on that this means:
    x is an element of A iffy x is an element of the empty set.
    How does on go from the first statement to the second? (Lemma 3.1.6)

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    1. I’m not quite sure where your confusion is, so this might take multiple rounds of comments to sort out. But here’s a first attempt at explaining this: He showed that the statement x \in A is false, and we also know that x \in \emptyset is false. Since both statements are false, they have the same “truth value” of \mathrm{False}. The connective \iff means that both sides of it have the same truth value, so we know \mathrm{False} \iff \mathrm{False}. But that’s just the same as saying x \in A \iff x \in \emptyset, since we know both of the individual statements evaluates to \mathrm{False}. (See Appendix A.1 for more about this \iff “if and only if” connective.)

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    2. Dear Mrs.Rice,
      Thanks for the swift reply.I understand now.I was trying to convert the iff statement in its logical equivalent,with AND and OR operators,and did not succeed in matching this with the original statement:x not an element of A AND
      x not an element of the empty set.

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    3. Dear Mr.Rice.I want to apologise for mistakingly calling you Mrs.Rice.You see,in my country,most names ending in a are female names.When I looked you up,I saw you are male.Sorry!

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  50. Do exercise 5.4.9 so that Exercise 6.1.8 (h) doesn’t need the redundant proof of min(x, y) = -max(-x, -y). I have 4th edition

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