# New research and re-discovering classic results in “basic” real analysis

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?

• 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! – Victor Dec 31 '14 at 11:04
• 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) – Victor Dec 31 '14 at 19:07
• @Victor : Perhaps you should post an answer with more details about your nice example. – Timothy Chow Dec 31 '14 at 21:28
• I don't think bounties are appropriate for "community wiki" questions – Yemon Choi Jan 3 '15 at 12:15
• @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 – Yemon Choi Jan 3 '15 at 12:32