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?
Best Answer
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.