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Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus).

This article is an example.

I would like to collect a "big list" of such novel proofs of old results. Note, however, that I am only looking for proofs that represent an improvement (in some sense) over standard alternatives which can be found on most textbooks.

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  • $\begingroup$ Do proofs avoiding assumption of certain postulates, like excluded middle, count as improvement to you? If so, there are surely many of those not usually provided in textbooks. $\endgroup$ – Nikolaj-K Dec 11 '14 at 21:16
  • $\begingroup$ @NikolajK I don't quite understand what you mean: I am courious to see an example of such a proof. $\endgroup$ – Dal Dec 11 '14 at 21:18
  • $\begingroup$ Constructive math is cautious about it's tools and reproves many theorems. Also, some theorems don't go through (as viewed form the constructive perspective) and are substituted by variations, e.g. this. The are improvement in that they actually contain algorithms for forming this and that object. See also this talk. It also reminds me of Reverse mathematics. $\endgroup$ – Nikolaj-K Dec 11 '14 at 21:27
  • $\begingroup$ @NikolajK, I didn't really think of this when I said "novel". Still, I think that this is very interesting: I will surely watch the talk. Thank you. $\endgroup$ – Dal Dec 11 '14 at 21:32
  • $\begingroup$ Similar question: math.stackexchange.com/questions/1056291/… $\endgroup$ – Victor Wang Dec 12 '14 at 7:08
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Andrew Bruckner's survey paper Current trends in differentiation theory includes a lot of examples of new simple proofs of results previously having difficult proofs, but most of the examples are probably past the level you want. Probably more appropriate would be the use of full covers in real analysis.

(ADDED NEXT DAY) When I got home last night I realized that the Bruckner paper I was thinking about isn't the paper I cited above, but rather the paper below. I couldn't find a copy on the internet, but most university libraries (at least in the U.S.) should carry the journal. Nonetheless, the use of full covers in real analysis, which I've already mentioned, is about as close a fit to what you're looking for as I suspect you'll get.

Andrew M. Bruckner, Some new simple proofs of old difficult theorems, Real Analysis Exchange 9 #1 (1983-1984), 63-78. [Go here for the zbMATH review (Zbl 569.26007) of the paper.]

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    $\begingroup$ What are the results re-proved by Bruckner? $\endgroup$ – Dal Dec 11 '14 at 21:36
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    $\begingroup$ I don't have access to Bruckner's paper where I'm at, but googling its title along with my name brought up a couple of stackexchange answers of mine that mention results in the paper. One is Importance of a result in measure theory and the other is Construction of a Borel set with positive but not full measure in each interval. $\endgroup$ – Dave L. Renfro Dec 11 '14 at 21:40
  • $\begingroup$ $+1$ Thank you for the reference. Do you know any other examples of new proofs of classic theorem in calculus? $\endgroup$ – Dal Dec 16 '14 at 16:49
  • $\begingroup$ I'll keep my eye open for such results and come back here to include any that I find in the future. However, off-hand (i.e. without possibly extensive searching through my stuff at home) I don't know of anything else. $\endgroup$ – Dave L. Renfro Dec 16 '14 at 18:42
  • $\begingroup$ Thank you very much for being so helpful :). $\endgroup$ – Dal Dec 16 '14 at 21:21

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