Talk:Riemann Removable Singularities Theorem

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I appreciate that (c) may well imply (d) from some definition of bounded, but as we have not defined bounded or neighborhood in terms of the $o$ function (I'm still not sure how it follows), we need to make sure this is specified.

I also note the statement that "a continuous function is locally bounded" but I'm not sure we've got this up yet as a proof. And if we have, we need to provide a link to it.

I've been doing what I can filling in the boring technical detail to build up these results from scratch, but I haven't been able to get round to everything yet. There are still loads of gaps. Complex analysis is an area I haven't really started on yet, I'm battling through (shakily) metric spaces and topological spaces with a view to providing some results sufficiently general that we won't need to provide too many detailed technical proofs - but as I say, this all still needs to be filled. --prime mover (talk) 22:54, 15 February 2009 (UTC)

There is always a tension in mathematical writing between the level of detail and keeping proofs understandable. The idea of giving every single detail makes it essentially impossible to present more complicated results in a way that anyone will ever read. Indeed, people likely to read these proofs would be expected to fill in minor details themselves - I do not think that any complex analysis textbook would give the details mentioned above.

(Another issue is the complete formulization of mathematical proofs, which is also a worthwhile undertaking. However, I believe it complements rather than replaces the idea of writing proofs that have the ideas clearly presented and are readable by humans.)

Of course, electronic media have other opportunities than print, and it could be possible to exploit this. What I mean is that, within a proof, if there is a statement "clearly ...", this statement could have a link that provides additional details. Thus, a reader who understands this point (or is willing to accept it at least for the time being) can read on easily, while those who would like to see the details can follow the link. Also, if someone writes the overall proof, then such little details can be filled in by others, without encumbering the full flow of the argument.

I am not entirely sure how easy it would be to implement this with wiki technology, without creating a plethora of additional pages. Are subpages (and sub-sub-pages etc.) supported? I believe this is a discussion worth having, and perhaps this comment should be moved to another, more public thread. If ProofWiki intends to grow to include much of classical mathematics (not to mention contemporary results), these questions should be addressed sooner rather than later.

As for the mathematics in question, by the definition of the $o$-Notation, the claim means that $|f(z)|/|z-z_0|\to 0$ as $z\to z_0$, if $f$ is bounded near $z_0$. This is trivial because the product of a bounded sequence and a null sequence is again a null sequence. lasserempe 08:51, 16 February 2009 (UTC)