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    I did this a while ago and now looking back at my working out, something doesn't feel right so I just want to check if my working out is right.

    let a_n = \frac{1}{n^2} and a_m = \frac{1}{m^2} and let  \epsilon > 0 be given.

     |a_n - a_m| < \epsilon \Rightarrow |\frac{1}{n^2} - \frac{1}{m^2}| < \epsilon

    Since  |\frac{1}{n^2} - \frac{1}{m^2}| < \frac{1}{n^2} + \frac{1}{m^2}, let  \frac{1}{n^2} + \frac{1}{m^2} < \epsilon

    This occurs when \frac{1}{n^2} < \frac{\epsilon}{2} and  \frac{1}{m^2} < \frac{\epsilon}{2}

    \Rightarrow n > \sqrt\frac{2}{\epsilon}

    So, if n is an integer larger than  n > \sqrt\frac{2}{\epsilon} then,  m, n > N \Rightarrow  |a_n - a_m| < \epsilon
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    It seems right but your wording is quite clumsy... and I think you got your lower-case n and upper-case N mixed up at some point towards the end too.

    To make it clearer, just say that if N > \sqrt{\frac{2}{\varepsilon}}, then if n>N then \dfrac{1}{n^2} < \dfrac{\varepsilon}{2}, and so for all m,n > N, |a_m - a_n| \le \dfrac{1}{m^2} + \dfrac{1}{n^2} < \varepsilon.
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    It took me a while to have the realization that at university they're not looking for your working, they're looking for a proof. The working/thinking should be hidden away where no one can see it.

    In this case the working you hide is how you figured out the correct N. Instead, just stick it in and show it works. So, just trudge through the definition of a Cauchy sequence, showing that it holds for the N you've figured out.
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    (Original post by JoMo1)
    It took me a while to have the realization that at university they're not looking for your working, they're looking for a proof. The working/thinking should be hidden away where no one can see it.

    In this case the working you hide is how you figured out the correct N. Instead, just stick it in and show it works. So, just trudge through the definition of a Cauchy sequence, showing that it holds for the N you've figured out.
    Really? That seems so counter-intuitive to me as in I feel like I'd lose marks for missing information. I get what you're saying but the proof just feels less rigorous (to me) somehow if I don't show it all from each step to the next step.
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    (Original post by Preeka)
    Really? That seems so counter-intuitive to me as in I feel like I'd lose marks for missing information. I get what you're saying but the proof just feels less rigorous (to me) somehow if I don't show it all from each step to the next step.
    I really don't see anything wrong with doing it the way you did it, as long as your arrows end up pointing in the right direction in the end. That is, instead of just saying \text{(stuff)} < \varepsilon \implies N > \text{(other stuff)}, you should say N > \text{(stuff) and } m,n > N \implies |a_m-a_n| < \varepsilon... but you did that, so it's fine.

    Hiding your working out isn't a good idea if you're unsure what can or can't be hidden; I mean, putting all your working out as part of your proof might make it look inelegant, but if it's right then that's not really a problem. [You won't lose marks either way unless you cut out vital steps.]
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    (Original post by nuodai)
    I really don't see anything wrong with doing it the way you did it, as long as your arrows end up pointing in the right direction in the end. That is, instead of just saying \text{(stuff)} < \varepsilon \implies N > \text{(other stuff)}, you should say N > \text{(stuff) and } m,n > N \implies |a_m-a_n| < \varepsilon... but you did that, so it's fine.

    Hiding your working out isn't a good idea if you're unsure what can or can't be hidden; I mean, putting all your working out as part of your proof might make it look inelegant, but if it's right then that's not really a problem. [You won't lose marks either way unless you cut out vital steps.]
    Yeah I suppose I will keep it as it is because as you said I wouldn't be sure if I would be cutting out steps that they are looking for. thank you for your help
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    You can cut out any steps that you only needed to find a value of N that works. You can't cut any steps that are required to show your value of N works.

    A lot of people actually 'cheat' in a way similar to what JoMo1 suggests.

    You see, these proofs all follow a basic template:

    "Take \epsilon > 0. Let N = magic. Take m, n > N. Then ... {whatever calculations you need to show your value for N works}".

    So, what you do is leave a gap for magic (since you don't know what it is yet). Then, do your calculations. As you do them, you'll find what N has to be. Then go back and fill that in.

    In terms of producing a quick, simple proof, this works very well. The logic flows all in one direction (in your original proof, you start from the desired conclusion, work backwards to find what value of N works, and then argue forwards again to say "and it really does work for this value of N"), which makes it a bit easier to read. But, it does hide where your value of N came from a bit. Not really a problem as an undergrad, but not so great if you're teaching.
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    (Original post by nuodai)
    I really don't see anything wrong with doing it the way you did it, as long as your arrows end up pointing in the right direction in the end. That is, instead of just saying \text{(stuff)} < \varepsilon \implies N > \text{(other stuff)}, you should say N > \text{(stuff) and } m,n > N \implies |a_m-a_n| < \varepsilon... but you did that, so it's fine.

    Hiding your working out isn't a good idea if you're unsure what can or can't be hidden; I mean, putting all your working out as part of your proof might make it look inelegant, but if it's right then that's not really a problem. [You won't lose marks either way unless you cut out vital steps.]
    I would say that anyone studying what appears to be a university analysis course should get to grips with what is and isn't required with epsilon-(delta/N) style proofs. Answers that are a stream of consciousness do tend to make your proofs messy and more inconsistent than writing them out in a more conventional way (hence the convention) and will make it easier for the examiner to say you did it wrong or for you to make a genuine mistake.

    At Oxford you also get marked on 'style', something like 10% of the exam in 2nd year. I don't know how it works other places, but at university level I'd expect convoluted logic to lose some marks whether fairly or unfairly.


    (Original post by DFranklin)
    In terms of producing a quick, simple proof, this works very well. The logic flows all in one direction (in your original proof, you start from the desired conclusion, work backwards to find what value of N works, and then argue forwards again to say "and it really does work for this value of N"), which makes it a bit easier to read. But, it does hide where your value of N came from a bit. Not really a problem as an undergrad, but not so great if you're teaching.
    In lecture notes we've had a neat proof presented and then a much less rigorous, 'thinking behind the proof' argument given afterwards, which tends to work quite well. If you're teaching live actually pointing out that you're leaving a blank to fill in later tends to give you an idea of how to actually go about doing it yourself. At the root of it, conventional proof-writing style exists for a reason and in the first year or two, writing rigorous maths is more difficult than understanding the maths, and me learning that the hard way has made me a tad pedantic about it now.
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    (Original post by JoMo1)
    At Oxford you also get marked on 'style', something like 10% of the exam in 2nd year. I don't know how it works other places, but at university level I'd expect convoluted logic to lose some marks whether fairly or unfairly.
    At all 3 unis I've attended, you wouldn't get marked down for convoluted logic (as long as it's right); Oxford would seem a bit unusual in this respect. (And I'd be interested to know just how "style" is assessed - it's obviously rather subjective).

    One thing I did hear back from an examiner was that when you set things out 'properly', you're more likely to get the benefit of the doubt if you do something a little dodgy (as much as anything because the examiner is more likely to fall into "autopilot mode" thinking "yeah, OK, yeah, OK, full marks").

    I'd also say, (and a fairly high profile mathematician said the same thing, although I can't remember who right now - might have been Korner) that the approach where you "hide all the failed approaches, ugly calculations, and just present the finished proof as if "take N = e^{1/\epsilon^2} + 913" was the most obvious thing in the world" isn't actually a very helpful way of presenting many of these topics.

    A particular example I remember: there would usually be a question where you had to decide whether a certain set was countable. And these questions can usually be trivialised by making aggressive use of the fundamental theorem of arithmetic. [e.g. the set S of all finite subsets of N is countable. Proof: let p_n be an enumeration of the primes; given a finite subset A={a_1, ..., a_m} define f(A) = \prod p_i^{a_i}, then f is an injection into N, thus S is countable].

    It's neat, and very efficient in an exam, but if that's the only technique you know you're missing a lot of material. (I should perhaps confess it was pretty much the only technique I knew - but then I didn't actually go to the lectures for that course).

    Edit: Have you read Littlewood's Mathematician's Miscellany? If you're a stickler for rigour, it's somewhat interesting to see his thoughts about it. [Roughly - not as important as ideas - any decent mathematician can put in the rigour after if needed. I was a little surprised].
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    (Original post by DFranklin)
    At all 3 unis I've attended, you wouldn't get marked down for convoluted logic (as long as it's right); Oxford would seem a bit unusual in this respect. (And I'd be interested to know just how "style" is assessed - it's obviously rather subjective).

    One thing I did hear back from an examiner was that when you set things out 'properly', you're more likely to get the benefit of the doubt if you do something a little dodgy (as much as anything because the examiner is more likely to fall into "autopilot mode" thinking "yeah, OK, yeah, OK, full marks").

    I'd also say, (and a fairly high profile mathematician said the same thing, although I can't remember who right now - might have been Korner) that the approach where you "hide all the failed approaches, ugly calculations, and just present the finished proof as if "take N = e^{1/\epsilon^2} + 913" was the most obvious thing in the world" isn't actually a very helpful way of presenting many of these topics.

    A particular example I remember: there would usually be a question where you had to decide whether a certain set was countable. And these questions can usually be trivialised by making aggressive use of the fundamental theorem of arithmetic. [e.g. the set S of all finite subsets of N is countable. Proof: let p_n be an enumeration of the primes; given a finite subset A={a_1, ..., a_m} define f(A) = \prod p_i^{a_i}, then f is an injection into N, thus S is countable].

    It's neat, and very efficient in an exam, but if that's the only technique you know you're missing a lot of material. (I should perhaps confess it was pretty much the only technique I knew - but then I didn't actually go to the lectures for that course).

    Edit: Have you read Littlewood's Mathematician's Miscellany? If you're a stickler for rigour, it's somewhat interesting to see his thoughts about it. [Roughly - not as important as ideas - any decent mathematician can put in the rigour after if needed. I was a little surprised].
    I'd agree with pretty much all of that, especially the Littlewood bit. Ideas are difficult, the rigour is easy. That's why first year courses tend to be relatively easy conceptually so that you can focus on getting that rigour sorted. If you don't take the opportunity to get it sorted then, you're going to run into trouble later.

    I'm also not suggesting that the OP changes their thought pattern, or that it should be taught in the format I suggested they write it in. I'm suggesting what is essentially exam technique: you're trying to convince the examiner/reader that you're argument is foolproof. The original argument didn't feel like a nice, confident piece of university analysis and I was suggesting how it could be improved to look as such.
 
 
 
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