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Sometimes, it happens that researchers publish a new proof of an old well-known result in "basic real analysis" (I'm referring to what some American people may call "honors calculus"). For instance, we can consider this article.

I have two questions:

(1) What are some examples recent novel proofs of old well-known results in "basic real analysis"?

(2) Has it ever happened in recent times that such a proof had been particularly useful bringing about new insights into major problems?

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    $\begingroup$ My goodness! After reading this post i thought about Lars Olsen who would love this type of things, and the author in the link is him! $\endgroup$
    – Victor
    Commented Dec 31, 2014 at 11:04
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    $\begingroup$ never mind what others vote for. I have a nice example for you! The old method to prove $\pi$ is not algebraic is involved. But there is an elementary half-page proof using $\Gamma(\frac{1}{2})\times\Gamma(\frac{1}{2})$ and analytic tricks to deal with product of two integrals (argument is going from the contrary) $\endgroup$
    – Victor
    Commented Dec 31, 2014 at 19:07
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    $\begingroup$ @Victor : Perhaps you should post an answer with more details about your nice example. $\endgroup$
    – Timothy Chow
    Commented Dec 31, 2014 at 21:28
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    $\begingroup$ I do not favour closing this question. I find the second clause of the question particularly interesting: I would not find it hard to list some examples of novel proofs of classical, elementary results (not necessarily in analysis), but I find it much harder to think of examples such that the new proof has had substantial research consequences. I would be very interested to learn about such examples. $\endgroup$
    – Ian Morris
    Commented Jan 1, 2015 at 21:12

2 Answers 2

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Using Google Scholar to search for recent American Mathematical Monthly articles containing the term "new proof" turns up some candidates. For example, Steve Roman's paper on The Formula of Faà di Bruno derives the formula using the umbral calculus. The umbral calculus is a classical technique that has been revived to produce numerous interesting new results; I'm most familiar with applications in combinatorics, as explained in Ira Gessel's paper, but there are probably others.

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  • $\begingroup$ This is really interesting. Thank you. $\endgroup$
    – user60665
    Commented Jan 1, 2015 at 0:39
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Here is one I once saved:

It gave rise to two postscripts:

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