Subject 01
Mathematical Analysis I
Analiza Matematyczna I.1
The rigorous construction of the real continuum, limits, series, continuity, differentiation, and one-dimensional convexity.
A
Number Systems & Dedekind Completeness
Axiom of Completeness
Compare the Dedekind-cut formulation with the Cauchy-sequence completion of ℚ.
Kuratowski Pairs & Peano Axioms
Derive ℕ from Peano arithmetic, construct ℤ and ℚ through equivalence classes of ordered pairs, and establish the Archimedean property.
Supremum / Infimum Infrastructure
Study dense subsets of ℝ, accumulation points, open and closed intervals in metric topology, and formal irrationality proofs, including the irrationality of e.
B
Sequences & Discrete Limits
Stolz–Cesàro Theorem
Use the discrete analogue of L’Hôpital’s rule to resolve indeterminate limits of ratios of divergent sequences.
Cauchy Criterion for Sequences
Work with complete metric spaces; prove every convergent sequence is Cauchy and that ℝ is complete.
Subsequences & Accumulation
Develop lim sup and lim inf algebra, and prove the Bolzano–Weierstrass theorem for bounded real sequences.
Elementary Transcendentals via Limits
Define e = limₙ→∞(1 + 1/n)ⁿ and prove strict monotonicity and boundedness.
C
Infinite Series of Real and Complex Numbers
Szeregi Liczbowe
Positive Series
Comparison and limit-comparison tests; Cauchy condensation; d’Alembert ratio and Cauchy n-th root tests; Raabe and Kummer tests for borderline cases.
Arbitrary and Alternating Series
Leibniz criterion, Abel and Dirichlet tests, absolute versus conditional convergence, and the distinction between convergence modes.
Riemann Rearrangement Theorem
Prove that a conditionally convergent series can be reordered to reach any real sum or to diverge.
Products of Series
Study the Cauchy product of two series and Mertens’ theorem.
D
Functions, Continuity & Compactness
Heine and Cauchy Limits
Prove equivalence between the sequential definition of a function limit and the ε–δ metric definition.
Darboux Property
Prove the Intermediate Value Theorem through connectedness and apply it to root-existence arguments.
Weierstrass Extreme Value Theorem
Use sequential compactness to prove that continuous functions on [a,b] attain global extrema.
Uniform Continuity
Prove Heine–Cantor and study modulus of continuity together with Lipschitz and Hölder bounds.
E
Rigorous Differentiation & Convexity
Exact Mean Value Theorems
Rolle, Lagrange, and Cauchy mean value theorems, followed by a rigorous proof of L’Hôpital’s rule.
Taylor Formula
Peano infinitesimal remainder, Lagrange derivative remainder, and Cauchy integral and differential remainder forms.
One-Dimensional Convex Analysis
Epigraph and secant-line definitions; derivative characterizations of convexity; Jensen’s inequality and the arithmetic–geometric–harmonic mean, Cauchy–Schwarz, Young, and Hölder inequalities.