TITLE: New research and re-discovering classic results in "basic" real analysis
QUESTION [13 upvotes]: 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?

REPLY [7 votes]: Here is one I once saved:

Peter D. Lax, Change of variables in multiple integrals (1999).

It gave rise to two postscripts:

Peter D. Lax, Change of variables in multiple integrals. II (2001),
Nikolai V. Ivanov, A differential forms perspective on the Lax proof of the change of variables formula (2005).