TITLE: Are there any standard analysis facts that can be proven or arrived only by means of non-archimedean extensions of reals and non-standard analysis?
QUESTION [9 upvotes]: I wonder whether non-standard analysis, non-archimedean extensions of reals such as surreal or hyperreal numbers can help us in obtaining standard-analytic results which are not possible to obtain by means of standard analysis? 

Finding limits in closed form which are impossible to find in standard analysis
Generalizing functions to the values where standard analysis does not provide any satisfactory method
Eliminating undetermined forms where it is impossible by standard-analysis means
Finding identities and relations between standard numbers and functions that are inpossible by standard means

Or non-standard analysis is just as powerful as epsilon-delta method in regard of standard numbers?
I am asking this because evidently complex numbers gave a lot of new powerful methods and results - from solving algebraic equations towards Fourier transforms, integral forms for finite differences, number theory etc.

REPLY [7 votes]: To respond to the title question "Are there any standard analysis facts that can be proven or arrived only by means of non-archimedean extensions of reals and non-standard analysis?", while it is true that in principle every NSA proof has a standard translation, in practice such translation may exponentially increase the complexity of the proof, so a real resercher, even equipped with a supercomputer, may well come across a result that can be proved using NSA but cannot in practice be proved in a traditional framework. There are certainly results in the literature that currently only have NSA proofs.  User Gerry Myerson asked for examples. Such examples are provided by the work of Renling Jin and coauthors in additive combinatorics, and that of Terence Tao and coauthors on approximate groups. Whoever can add a single layer of quantifiers and thereby eliminate the infinitesimals and/or ultraproducts from the proofs given by Jin and Tao will earn my admiration.