 The approaches pioneered by Robinson and Nelson do not allow us to prove results about the standard real numbers that cannot be proved using standard real analysis. However, these approaches do provide simpler—and, in some sense, more intuitive—proofs of many theorems of standard real analysis. (On the pedagogical benefits of non-standard analysis, see, for example, Keisler (1976)). And there are cases of results in real analysis that were first proven using non-standard real analysis (see, for example, Bernstein and Robinson (1966).) Moreover, these approaches clearly show that we do not need to adopt the \(\epsilon\)-\(\delta\) formalization of the notion of a limit in order to have access to a fully rigorous development of real analysis.