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?