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Chapter 1 Axioms of the Real Number System 1.1 Introductory Remarks: Wha t c onstitutes a proof ? One of the hurdles for a student encountering a rigorous calculus course for the first time, is what level of detai l is expected in a proof. If ever y state men t is completely justified, the proof that 1 + 1 = 2 takes over 200 pages of Prin cipia Math emat ics (Russe ll & Whit ehea d). So obviously, some things must be taken for granted. Just how much? Knowing the answer requires some sophistication...for instructor and student to agree on what is to be assumed and what is to be justified. Often, precisely what a proof is is stated explicitly. For the record Definition 1.1  A  proof  is a finite sequence of statements, each of which is an axiom, one of the hypotheses of the theorem, or follows from the preceding statements of the proof by elementary rules of inference, and the last statement of the proof is the conclusion of the theorem. At the end of this chapter is an example of detailed mathematical reasoning, beyond what we shall require in this text, but detailed enough to indicate to the reader what truths rest upon what assumptions. 1

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