Mathematical Analysis 1

1. The Real Field
Real numbers: ordering, operations, and properties. Bounded and unbounded sets. Upper and lower bounds, supremum and infimum, maximum and minimum. Intervals. Completeness of the real numbers. Density of rational and irrational numbers. Archimedean property and n-th roots. Decimal representation of rational numbers.

2. The Complex Field
Definition of the complex field and its operations. Isomorphism between ℝ and a subfield of ℂ. Algebraic, trigonometric, and exponential forms of complex numbers. Conjugate, modulus, powers, and roots. De Moivre's formula. Fundamental Theorem of Algebra.

3. Cardinality
Cardinality of sets. Equipotent sets. Countability of ℤ, ℚ, and operations on countable sets. Cardinality of the continuum and uncountability of ℝ. Power set and cardinality of (ℕ).

4. Sequences and Limits
Real sequences: convergence, divergence, oscillation. Uniqueness of limits, compactness theorems, Bolzano-Weierstrass theorem. Cauchy criteria. Limits on the extended real line, one-sided limits. Comparison theorems. Indeterminate forms. Monotone sequences, notable limits, the constant e, comparison of infinities and infinitesimals. Limsup, liminf, subsequences, and accumulation points

5. Numerical Series
Definition and convergence of series. Absolute convergence and convergence tests (comparison, root, ratio, condensation). Generalized harmonic series. Positive-term and alternating series. Brief notes on unconditional convergence.

6. Function Limits and Continuity
Limits of real functions, sequences, and compositions. Bounded functions and their extrema. Fundamental limit theorems. Asymptotes, infinitesimals, continuity and types of discontinuity. Monotonic functions and continuous inverses. Uniform continuity and Heine-Cantor theorem. Lipschitz functions and semicontinuity.

7. Differential Calculus (One Variable)
Derivatives and continuity. Non-differentiable points. Differentiation rules and derivatives of compositions and inverse functions. Sign of the derivative, local extrema. Fermat's, Rolle's, Lagrange's, and Cauchy's theorems. Higher-order derivatives, Taylor's formulas. Convexity and inflection points. Qualitative graph analysis.

8. Integral Calculus (Riemann)
Indefinite integral and main integration techniques. Riemann integrability, geometric meaning. Properties of the definite integral. Fundamental Theorem of Calculus. Integral functions and their derivatives. Improper integrals and convergence criteria. Qualitative analysis of functions defined by integrals.