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This question is related to the question posted here. Would a shorter proof to those in the answers, such as:

Take the subsequence $\{a_m\}$ of $\{a_n\}$ where $m > 0$. By induction on $m$ $$\forall{m}( a_m < a_{m+1})$$ Therefore $$ \forall{\epsilon} > 0 \ \exists{N} \ \forall{m_0,m_1} \ (m_0,m_1 \geq N \ \wedge \ m_0 \geq m_1 \ \ \Rightarrow \ \ m_0 - m_1 < \epsilon)$$ and $\{a_m\}$ is Cauchy. Hence it converges, so its parent sequence converges. Therefore its parent sequence is Cauchy.

be acceptable?

Furthermore, what is the minimum amount of detail required in a proof? I am curious because I am switching into a maths program, however all my maths is self taught. I am not used to having my proofs reviews or having to prove to anyone but myself.

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providence
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  • Why must $a_m<a_{m+1}$? And why must the subsequence be Cauchy? I don't think you've proved it. – Andrew Aug 17 '12 at 11:49
  • And I think by "$m_o-m_1<\epsilon$" you meant "$a_{m_o}-a_{m_1}<\epsilon$". – Andrew Aug 17 '12 at 11:51
  • @Andrew So I do need to go into detail in that area? Looking at the definition of the sequence it is clear, so I was not sure. – providence Aug 17 '12 at 11:57
  • @providence: consider the sequence $a_n=2^{-n}$. Does your proof hold in this case? – Dejan Govc Aug 17 '12 at 11:58
  • @DejanGovc It appears so to me. Provided n tends to positive infinity. – providence Aug 17 '12 at 12:06
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    @providence, in Dejan's example $(1/2,1/4,1/8,\ldots)$ we don't have $a_m<a_{m+1}$. Also, on second look, what do you mean by "the" subsequence ${a_m}$? There are many subsequences, and I'm not sure if you have a specific one in mind. – Andrew Aug 17 '12 at 12:06
  • @Andrew the subsequence where all the indices $n$ are positive integers. Edit: I see it does not apply to DejanGovc's case. Does this affect is applicability to the specific case in the question? Was my definition of the subsequence not clear enough? I apologise for the multitude of questions. – providence Aug 17 '12 at 12:08
  • @providence, ah, I thought maybe that was what you meant. And presumably ${a_n}$ begins with $a_0$? The trouble with induction on $m$ is that $m$ is an index running through all positive integers, rather than simply an integer. – Andrew Aug 17 '12 at 12:13
  • But if say, you try to do induction on the number of terms you cut off the beginning of the sequence, then the base case would not get easier. You still have to prove that the (sub)sequence is Cauchy, which doesn't depend at all on the first finitely many terms of the original sequence. – Andrew Aug 17 '12 at 12:17
  • @Andrew could I fix an x, say, and do induction on that? – providence Aug 17 '12 at 12:20
  • @providence, it's not clear to me induction will work -- I think you'll end up with the same problem. Your accepted solution for the previous question is nice. It is also a standard argument. Ultimately, you need to try to bound $|a_{n_i}-a_{n_j}|$ by a given $\epsilon$ for all $n_i,n_j$ large enough, and I don't think induction on subsequences helps with this. – Andrew Aug 17 '12 at 12:30
  • For the last question, I've once heard someone suggest that you should write down enough detail so that every step is obvious... and then add one more level of detail beyond that. –  Aug 17 '12 at 15:35

2 Answers2

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This answer solely regards the latter portion of the question.

Whenever you write a proof, it is vitally important that you know your audience. This is key because a proof is simply an argument intended to convince the reader that the purported statement is true (in general, only automatic theorem provers actually show all the details). So the question then becomes: "how much detail should I leave out?" This is precisely where your audience comes into play. For example, a professor of mathematics can, in general, make greater logical leaps than an undergraduate (for a variety of reasons such as familiarity with the material, etc). Thus the proofs presented to the undergraduate student would need to be more detailed than those presented to the other party.

However, there is one significant exception to the above rule: proofs written for coursework. In this case, the student is writing the proof for the professor to read, and so the professor could handle larger logical leaps. But, the student shouldn't write this way because the professor cannot tell if the student could actually fill in the details omitted. So, the student should write a proof as if he/she is communicating it to his/her classmates. Doing this should correspond to a level of detail where the audience (the professor) knows that the student has sufficient understanding of the material.

In summary, the level of detail required depends on the audience for whom you are writing.

J. Loreaux
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  • Pretty much exactly my thoughts. – tomasz Aug 17 '12 at 14:13
  • There is however, an exception to the exception: when the professor knows you well enough to know you can fill in the details, he might be more lenient in that department. But then again, it boils down to the “know you audience” (or, rather, “know your professor” :) ) part. – tomasz Aug 17 '12 at 14:16
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    I got a nice piece of advice from a professor last fall. He said that I should be able to explain any step, in words, in less than 30 seconds. For a refereed paper this is maybe overkill, but I found it was surprisingly good advice for assignments: it was easy to test on myself, asking, is this really true and can I quickly jot down an airtight explanation. – Andrew Aug 17 '12 at 18:38
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I can say the following about the required detail in proofs in math programs. Your teachers have to tell the difference between provers and posers. Therefore, your proofs should be quite detailed. You can skip anything that is explained in the lecture notes or in the books provided that you properly refer to them; besides that your can skip some high school stuff like standard limits, but not much else.

Wouter Stekelenburg
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