Keisler wrote a monograph to accompany the textbook: https://people.math.wisc.edu/~keisler/foundations.html https://people.math.wisc.edu/~keisler/foundations.h

Keisler wrote a monograph to accompany the textbook: https://people.math.wisc.edu/~keisler/foundations.html https://people.math.wisc.edu/~keisler/foundations.html It's a bit more than an instructor's manual. It covers the construction of the hyperreals in two ways: One (in section 1E) is relatively short, and is like the compactness theorem approach Abraham Robinson originally used in his book ("Non-standard Analysis", Princeton U. Press). The second construction uses ultraproducts (in Chapter 1*). Keisler gives a quick introduction to the logic needed to understand the ultraproduct construction, beginning with formal languages and sentential logic. So the prerequisites are there, and explained concisely but clearly. The ultraproduct construction comes up elsewhere; for applications to other areas of math, see Martin Davis's book "Applied Nonstandard Analysis" (Dover Publications). (There's a review of the books by Keisler, Davis, and Stroyan and Luxemburg: https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/S0002-9904-1978-14401-2.pdf https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978...). The rest of Keisler's monograph contains mostly nonstandard proofs of many of the results in the textbook. I like it a lot, and anyone who's interested in this stuff should grab the PDF. Another book that is short and moderately elementary is "Infinitesimal Calculus" by James Henle and Eugene Kleinberg, also published by Dover [which also publishes a paper version of Keisler's text, and is generally a great publishing company]. When Keisler's book came out, someone decided that it would be reviewed for the Bulletin of the AMS by Errett Bishop, who (as I recall) was a noted constructivist. You can read the review here: https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977-14264-X/S0002-9904-1977-14264-X.pdf https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977... Having a constructivist review a calc textbook that uses nonstandard analysis was probably not a great idea. The review ends with: "Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique." I believe Keisler wrote a reply in a subsequent issue of BAMS, but I can't find it.