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](http://www.jstor.org/discover/10.2307/4145046?uid=3738296&uid=2&uid=4&sid=21104830505661).


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?
> 
> 
> 



New research and re-discovering classic results in "basic" real analysis
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](http://www.jstor.org/stable/2320788) 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](http://arxiv.org/abs/math/0108121), but there are probably others.


