Course Syllabus
This syllabus outlines the modules of Real Analysis.
- Dedekind theory of real numbers, algebraic and order properties of $\mathbb{R}$, Archimedean property, density theorem, completeness property of $\mathbb{R}$, bounded sets, theorems on suprema and infima.
- Neighbourhood of a point in $\mathbb{R}$, open and closed sets, limit points and isolated points of a set, Bolzano-Weierstrass theorem for a set, derived set, closure and interior of a set.
- Sequence and its convergence, bounded sequence, monotone sequences, subsequences, limit of a sequence, limit theorem, Bolzano-Weierstrass theorem for sequences, limit superior and limit inferior, Cauchy sequence, Cauchy's general principle of convergence.
- Infinite series and their convergence, Cauchy criterion, tests for convergence (comparison test, D'Alembert ratio test, Cauchy root test, Raabe's test, logarithmic test, De Morgan and Bertrand test, Cauchy integral test, Cauchy condensation test, Gauss's test), alternating series, Leibniz test, absolute and conditional convergence.