# “Novel” proofs of “old” calculus theorems

Every once in a while some mathematicians publish (mostly on the American Mathematical Monthly) a new proof of an old (nowadays considered "basic") result in analysis (calculus).

I would like to collect a "big list" of such novel proofs of old results. Note, however, that I am only looking for proofs that represent an improvement (in some sense) over standard alternatives which can be found on most textbooks.

• Do proofs avoiding assumption of certain postulates, like excluded middle, count as improvement to you? If so, there are surely many of those not usually provided in textbooks. – Nikolaj-K Dec 11 '14 at 21:16
• @NikolajK I don't quite understand what you mean: I am courious to see an example of such a proof. – Dal Dec 11 '14 at 21:18
• Constructive math is cautious about it's tools and reproves many theorems. Also, some theorems don't go through (as viewed form the constructive perspective) and are substituted by variations, e.g. this. The are improvement in that they actually contain algorithms for forming this and that object. See also this talk. It also reminds me of Reverse mathematics. – Nikolaj-K Dec 11 '14 at 21:27
• @NikolajK, I didn't really think of this when I said "novel". Still, I think that this is very interesting: I will surely watch the talk. Thank you. – Dal Dec 11 '14 at 21:32
• Similar question: math.stackexchange.com/questions/1056291/… – Victor Wang Dec 12 '14 at 7:08

• $+1$ Thank you for the reference. Do you know any other examples of new proofs of classic theorem in calculus? – Dal Dec 16 '14 at 16:49