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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 Dec 31 '14 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 Dec 31 '14 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 Dec 31 '14 at 21:28
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    $\begingroup$ I don't think bounties are appropriate for "community wiki" questions $\endgroup$ – Yemon Choi Jan 3 '15 at 12:15
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    $\begingroup$ @Dal CW questions are those where there is no single correct answer, or those which are deliberately seeking a big list. In particular one does not usually accept a particular answer. If I understand the bounty system, then it will automatically select the highest voted answer, unless you selected an answer to accept. So the two mechanisms seem to go against each other $\endgroup$ – Yemon Choi Jan 3 '15 at 12:32
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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 Jan 1 '15 at 0:39
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Here is one I once saved:

It gave rise to two postscripts:

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