Real Analysis I

Syllabus

• Logic and proof techniques; classical inequalities; sets, functions, and sequences.
• Real numbers: axiomatic vs constructive foundations; decimal expansions; bounds, infimum/supremum; exponentiation.
• Limits of sequences (including divergence to infinity); limit laws; inequalities and limits; monotone sequences.
• liminf/limsup; Bolzano–Weierstrass theorem; Cauchy criterion for convergence.
• Countable sets (introductory results and examples).
• Global properties of real functions: monotonicity and convexity.
• Limits and continuity of functions; interior/accumulation/isolated points; transfer principles.
• Continuous functions on closed bounded intervals (key consequences); links between monotonicity, limits, and continuity; convexity and continuity.
• Core function classes: polynomials, rational functions, exponential/log/power functions, trigonometric and inverse trigonometric functions, hyperbolic functions.