Real analysis studies real numbers, real-valued functions, limits, continuity, sequences/series, convergence, and foundational theorems that underpin calculus and beyond.
The real numbers ℝ form a complete ordered field: every non-empty set bounded above has a least upper bound (supremum). This completeness distinguishes ℝ from ℚ and underlies many theorems in analysis.
Definition of a sequence: (aₙ). A sequence converges to limit L if for every ε>0, ∃N such that for all n≥N, |aₙ − L| < ε.
Limits of functions as x→a, one‑sided limits, and the ε–δ definition of continuity.
Infinite series ∑aₙ converges if the sequence of partial sums converges. Discussion of tests: comparison test, ratio test, absolute vs conditional convergence, etc.
Introduce continuity, uniform continuity, differentiability, Riemann (or intuitive) integration. Highlight key theorems like the Intermediate Value Theorem, Mean Value Theorem, etc. [oai_citation:1‡Wikipedia](https://en.wikipedia.org/wiki/List_of_real_analysis_topics?utm_source=chatgpt.com)
- Intermediate Value Theorem
- Mean Value Theorem and Taylor’s theorem
- Bolzano–Weierstrass theorem, Heine–Borel theorem (in ℝⁿ context)
- Examples of pathological functions: continuous but nowhere differentiable, functions with convergent series but non‑absolute convergence, etc.