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My question is genuine, and I want to improve my understanding of the website with it.


Recently, I've come across this question, which seems to me clearly (no pun intended) unclear. My reasoning was as follows:

  1. There is no question.
  2. He says he "does not understand (...)", which is not what he is supposed to understand, since what he states after that is not the definition.
  3. The next phrase after that also doesn't make sense.

and proceeded by saying:

Okay, so we can explain the concept of a Cauchy sequence. What then?
What is the question we are supposed to answer, after addressing
that issue?

Also, I noted that the entire relevant body of the question (by that, I mean the part which was not referential only) was "adressed" by my points above.

There were no arguments in the post with respect to my reasoning. However, this does not necessarily mean that I'm right, and apparently the community did not agree, since the question has only 1 close vote (me, although I am not the downvote).

I see two reasons for the disagreement of the community: either I'm wrong, or people didn't close due to empathy.

If the latter is true, there is nothing I can do but lament.

However, if the former is true, I can learn something from this situation in order to improve my judgement of future posts.

Therefore, I ask: Is the question I linked unclear? Why/why not?

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    $\begingroup$ It seems pretty clear to me. There is a question. It's how a Cauchy sequence can converge to an irrational number. The fact that the question is in the title and not in the body is a bit tacky, but very clear. I think the question gives me a pretty clear of the OP's level of understanding. I think this also appealed to Michael Hardy, which imight be why he wrote (and why he could write) his nice answer. I would add that you might comment to the OP about any unclarity you perceive, or read the existing answers to see if you're missing something. $\endgroup$ – davidlowryduda Jan 13 '16 at 4:28
  • $\begingroup$ You say There is no question. This is incorrect. The original post had a question in the title ("How can a Cauchy sequence converge to an irrational number?") and a question towards the end ("...how can the distance ever be less than the smallest real number or infinitesimal as the distance can never become 0.") albeit without a question mark. $\endgroup$ – Joel Reyes Noche Jan 13 '16 at 4:36
  • $\begingroup$ I understand that if we take the approach that "the question is the title", the question is clear. I agree with you there. However, it appears to me that the context (quite paradoxically) makes it unclear. The answer seems to be an evidence for it: it answers the question in essentially one line: "Take the sequence $x_n=\pi$". Question answered. What happens after that is the deconstruction of the misconceptions of OP, which goes a way that almost seems another question altogether. Summarizing, I believe the following is the case: (...) $\endgroup$ – Aloizio Macedo Jan 13 '16 at 4:45
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    $\begingroup$ The question "How can a Cauchy sequence converge to a irrational number?" is clear. The question "How can a Cauchy sequence converge to a irrational number, if the distances are not less than all positive numbers? etc etc" is unclear. As an analogy, I would present this: The question "How can a circle be a $1$-manifold?" is clear. The question "How can a circle be a $1$-manifold, if it comes back to itself?" is unclear. Do you think my analogy is incorrect? Or it is correct, and all questions are indeed clear? $\endgroup$ – Aloizio Macedo Jan 13 '16 at 4:45
  • $\begingroup$ I have retagged your post (specific-question), since it fits the way I understood your question. If you used the linked question merely as an example and you want in fact discuss a more general issue, then this tag should not be used (see tag-info). So if that is the case, please remove this tag. $\endgroup$ – Martin Sleziak Jan 13 '16 at 7:56
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    $\begingroup$ A lot of people come here with confused questions about limits and/or infinity. This one is actually one of the better ones I've seen. $\endgroup$ – Thomas Andrews Jan 13 '16 at 13:38
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    $\begingroup$ In particular, I think we have to expect some types of confusion in questions about certain topics. Quite a lot of us recall deep confusion at the beginning about "different sizes of infinity" and limits and the like. Sometimes the question is best closed because it is a duplicate - we have gotten an awful lot of these. $\endgroup$ – Thomas Andrews Jan 13 '16 at 13:55
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I actually think the post you link to is a pretty good question. It gives a clear question of:

How can a Cauchy sequence converge to an irrational number?

and then goes on to discuss the sequence $3,\,3.1,\,3.14,\ldots$ and express doubt that that's a Cauchy sequence. Then, even better, we get to see precisely where the OP's logic veers off course - and we even get the secondary question of

Does this definition of completeness apply where $\varepsilon$ is infinitesimal?

which is really in the same vein of the first (e.g. it asks, "Why did my thoughts about this Cauchy sequence converging to an irrational not work?") but more specific. All of this makes the question easy to answer well - we can see exactly what's confusing the OP and provide an answer that addresses that. This is to say that questions which lack rigorous meaning (or are rigorously nonsensical) aren't necessarily unclear.

That said, this question is also a good example of why context is helpful. For instance, if we boiled it down to something like:

Title: How can a Cauchy sequence converge to an irrational number?

I've been told that the sequence $3,\,3.1,\,3.14,\ldots$ is such an example, but it's not a Cauchy sequence.

then closing as unclear would make sense, since the only logical thing to do would be to ask the OP, "Why don't you think it's a Cauchy sequence?" - which is not an answer to their question. (And that question definitely doesn't apply to the question in question)

The issue of mathematically problematic questions has been discussed in another meta thread.

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  • $\begingroup$ First of all, thanks for the answer. I really appreciate it. I will not be able to read it carefully now, since it's late where I live, but I will tomorrow. However, having read it superficially, it seems that your answer resonates with this question. Is this the case? Please tell me, so I can also read the discussion in this other post and try to connect the dots in this issue. $\endgroup$ – Aloizio Macedo Jan 13 '16 at 5:07
  • $\begingroup$ @AloizioMacedo I'd say my answer here is much in line with that other post. $\endgroup$ – Milo Brandt Jan 13 '16 at 5:10

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