Course description:

This course provides a rigorous introduction to real analysis, covering both the foundations of the subject and its central theorems. Emphasis is placed on understanding concepts through proofs, illustrative examples, and problem-solving practice.

Topics

• Foundations
  • Basic logical concepts and proof techniques
  • The real numbers and their structure, Axioms of the real numbers
  • Subsets of R: boundedness, supremum/infimum, density, countability and uncountability
• Numerical Sequences & Series
  • Convergence of sequences: accumulation points, limsup/liminf
  • Bolzano–Weierstrass theorem, Cauchy criterion
  • Limit of recursive sequences
  • Asymptotic equality of sequences
  • Convergence tests and criteria
  • Infinite series: absolute vs. conditional convergence
• Functions of One Variable
  • Limit, limsup, liminf, oscillation, continuity and types of discontinuities
  • Uniform continuity (Heine–Cantor theorem)
  • Differentiation: mean value theorems, applications
  • Integration: Riemann integral, improper integrals, Lebesgue’s criterion, Riemann–Stieltjes integral
• Functions of Several Variables
  • Limits and continuity in higher dimensions
  • Differentiation and the chain rule
  • Extrema
• Sequences & Series of Functions
  • Pointwise vs. uniform convergence
  • Power series and Taylor series
  • Analytic functions
• Metric Spaces
  • Open/closed sets, convergence, compactness, connectedness
  • Completeness, separability
  • Banach fixed-point theorem

  The course balances rigorous theory with problem-solving practice. By the end, students will have a deep understanding of the building blocks of modern analysis.