Chapter 1: function the absolute value of a real number the two (most) important properties of the absolute value the two steps for a proof by induction upper bound, lower bound the maximum of a set, the minimum of a set bounded above, bounded below least upper bound, greatest lower bound the least upper bound property the (three) important properties of the real numbers the Archimedean property of the real numbers (property (i) from 1.4.2) the density property of the rationals in the reals one-to-one (injective) onto (surjective) one-to-one correspondence (bijective) cardinality the power set of A Chapter 2: sequence convergence of a sequence a sequence is convergent a sequence is divergent a sequence is bounded a sequence is increasing a sequence is decreasing a sequence is monotone the n'th partial sum of a series convergence of an infinite series subsequence Statement of the Bolzano-Weierstrass theorem Cauchy sequence The Cauchy criterion for series A series converges absolutely A series converges conditionally Absolute convergence test Comparison test Alternating series test a rearrangement of a series Chapter 3: an open set a limit point of a set an isolated point a closed set the closure of a set the complement of a subset of R a bounded set An open cover of a set A finite subcover of an open cover The sequential definition of compactness The open cover definition of compactness an interval two sets are separated a set is disconnected a set is connected Chapter 4: functional limit continuity at a point continuity on the domain uniform continuity statement of the intermediate value theorem Chapter 5: differentiability at a point, and the derivative at a point differentiability on the domain statement of the mean value theorem