Università Roma Tre

1. Sets. Definition. Logic quantifiers. Fundamental properties. Cartesian product. Binary operations; order relations; equivalence relations; functions.
2. Real numbers. The real number system: algebraic axioms and the supremum axiom. Natural numbers; induction principle and basic properties. The rational numbers are a field; Archimedean property; density of the rational. Roots and irrational numbers. Properties of roots.
3. Sequences and series.
Sequencies. (As functions from N to R). Limit of a sequence. Operations with limits. Limits involving ± ∞. Comparison theorem. Theorem of the permanence of the sign. Limits of monotone sequences. The Euler number e. Powers with real exponents and logarithms.
Numerical series. Convergence criteria for positive series. Absolute convergence. Alternating series (Leibnitz criterion). The exponential series. Irrationality of e. Hyperbolic functions. Definition series of trigonometric functions. Series double and the theorem of addition for the cosine. Definition of π. Properties of sine and cosine.
General theory of sequencies. Maximum and minimum limits. Sequencies and topology of the line. Cauchy sequences. Bolzano-Weierstrass Theorem. Countable sets. R is not countable.
4. Real functions. Limits of functions. Theorem of the permanence of the sign. Continuous functions. Composition of functions. Fundamental theorems on continuous functions (Weierstrass, theorem of zeros, preimage of open intervals and image of compact sets). Restrictions; left / right limits. Points of discontinuity. Continuous functions invertible / monotonous. The uniform continuity.

Rudin, W: Principles of Mathematical Analysis, McGraw-Hill, 1976