Real Analysis B.Sc. 2nd Year | Theorems related to Limit points and Closed sets | Proof of Theorems

Download mobile applications for complete videos in the future https://bit.ly/3bT59Wl Join What's App Group for math students https://chat.whatsapp.com/IhVrtNoFzXt... Join Telegram Channel for notes & pdf https://t.me/mathsbylalitsir In this series, we will study about The set of real numbers as a complete ordered field, incompleteness of Q, Archimedean and denseness properties of R, Modulus, Intervals, Definition of a sequence, Theorems on the limit of the sequence bounded and monotonic sequences, nested interval theorem, Cauchy’s sequence, Cauchy’s convergence criterion. Convergence of series of non-negative terms, their various tests (Comparison; D’Alembert’s ratio, Cauchy’s nth root, Raabe’s, Gauss, Logarithmic, Demorgan and Bertrand's, Cauchy’s condensation, proof of tests not required) for convergence, Alternating series, Leibnitz’s test, Series of arbitrary terms, absolute and conditional convergence, Abel’s and Dirichlet’s tests. Equivalent sets. Finite and infinite sets denumerable sets, Countable and uncountable sets. Interior point of a set, open set, limit point of a set, Bolzano-Weierstrass theorem. Closed set. Dense in itself and perfect sets. Cantor’s ternary set. Definition of the limit of a function. Continuity of a function - Cauchy’s and Heine’s definitions with their equivalence. Types of discontinuities. Properties of continuous functions are defined on closed intervals. Uniform continuity. Differentiability, Rolle’s theorem, Lagrange’s and Cauchy’s mean value theorems, and their geometrical interpretations. Taylor’s theorem with various forms of remainders. Darboux’s intermediate value theorem for derivatives. Darboux sums and their properties. Riemann integral, Integrability of continuous and monotonic functions. Mean value theorems of integral calculus, The fundamental theorem of integral calculus. Improper integrals and their convergence comparison tests. Abel’s and Dirichlet’s tests. #Lalitsir #BSCmaths #Onlineeducation #KotaUniversityBScMaths #MdsUniversityBScMaths #mathsbylalitsir #bscmaths #engineeringmaths #uniquemathsDownload mobile applications for complete videos in the future https://bit.ly/3bT59Wl