A Little Nonstandard Analysis
For my first blog post we'll look at a little interesting nonstandard analysis (NSA) problem that allows me to provide a non-geometric proof of the following: \[\lim\limits_{x \rightarrow 0} \frac{\sin x}{x} = 1\] In NSA terms we would write: \[\frac{\sin \epsilon}{\epsilon} \approx 1\] for all nonzero infinitesimals \(\epsilon\). For this problem we will need three things: Some axioms to start, the standard part definition, and a few fairly basic tools. Informally we'll describe NSA the way Isaac Goldbring does in his lecture notes (pg. 4) with some light edits: (NS1) \((\mathbb{R}; +, \cdot, 0,1,<)\) is an ordered subfield of \((\mathbb{R}^*; +, \cdot, 0, 1, <)\). (NS2) \(\mathbb{R}^*\) has a positive infinitesimal element, that is, there is \(\epsilon \in \mathbb{R}^*\) such that \(\epsilon > 0\) but \(\epsilon < r\) for every \(r \in \mathbb{R}^+\). (NS3) For every \(n \in \mathbb{N}\) and every \(\mathbb{R}^n \xright...