Real analysis
Undergraduate · Math
Syllabus focus
Theoretical / proof-based
Pricing calculator
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$1,162 · Real analysis · 18 tutoring hrs
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Topics typically covered
Theoretical / proof-based
Real numbers and sequences
- Ordered fields and the axioms of R; least upper bound property
- Sequences of real numbers; limits, Cauchy sequences, and convergence
- Subsequences and the Bolzano–Weierstrass theorem
- Series of real numbers; absolute and conditional convergence
Topology and continuity
- Topology of R: open and closed sets, compactness, and the Heine–Borel theorem
- Limits of functions; sequential characterization of limits
- Continuity; extreme value and intermediate value theorems (with proofs)
- Uniform continuity and its distinction from pointwise continuity
Differentiation and integration
- Differentiation: definition, mean value theorem, and Taylor's theorem
- Riemann integration: upper and lower sums, integrability criteria
- Fundamental Theorem of Calculus in the rigorous setting
Advanced topics
- Sequences and series of functions; pointwise vs uniform convergence
- Power series and analytic functions (introduction)
- Metric spaces (optional capstone): definitions and examples
Notes
Topics reflect common real analysis / advanced calculus syllabi for mathematics majors at US universities. Strong calculus and proof-writing background is expected. Exact coverage varies between one-semester and two-semester sequences.