I´m learning some non-standard analysis. Quite some basic properties and theorems are "easier" to prove in the setting of non-standard analysis. But I´m searching for the converse, does anyone know some examples of results which are "more difficult" (the structure of the proof is more advanced, the proof is lengthier,...) to prove using non-standard analysis?
Many thanks!
The wording of this question suggests a common misconception that involves interpreting the term "non-standard" in the name of this field in its generic meaning. For this reason it is preferable to refer to the field as "Robinson's framework for analysis with infinitesimals", "infinitesimal analysis", or something along those lines. The point is that Robinson's framework is a conservative extension of the classical one and as such *incorporates all of the techniques available in the "classical" setting you referred to. In other words, Robinson's framework is part of the classical framework but provides new tools that can be used fruitfully where the old tools become too awkward. Thus by definition it is impossible to give "properties or theorems" that are harder to prove in Robinson's framework than in the "classical" framework as you put it.