The best book I have read on nonstandard analysis is "Lectures on the Hyperreals", by Goldblatt, which is <i>excellent</i>.<p>On the more practical end, Pétry's "Analyse Infinitésimale: une présentation non standard" is an extremely readable first-year-undergrad level textbook that shows you how simple first-year analysis is when approached this way (but it doesn't go much into what it actually <i>is</i>).<p>(Of course, per Thomas Forster IIRC, the purpose of a first-year course in real analysis is not actually to teach you about real analysis; so in my view this rather defeats the point. First-year real analysis is taught primarily to make sure you understand the difference between "for all x there exists y" and "there exists x such that for all y".)
IMO this is a good book on nonstandard analysis: <a href="https://eml.berkeley.edu/~anderson/Book.pdf" rel="nofollow">https://eml.berkeley.edu/~anderson/Book.pdf</a><p>Infinitesimal Methods in Mathematical Economics by Robert Anderson, math and econ professor at Berkeley
Terence Tao's blog entry "Nonstandard analysis as a completion of standard analysis"<p><a href="https://terrytao.wordpress.com/2010/11/27/nonstandard-analysis-as-a-completion-of-standard-analysis/" rel="nofollow">https://terrytao.wordpress.com/2010/11/27/nonstandard-analys...</a>