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• Introduction to real number systems ,concept of least upper bound and greatest lower bounds
• Definition of supremum , Infimum ,relevant problems, ordered field theorems and axioms
• Existence of Real field,Archimedean property,extended real number system.Schwar’z inequality and relevant questions
• Definition of sequence and its types,Bernoulli’s inequality,examples of increasing and decreasing sequences
• Bounded sequence,convergence of sequence,Cauchy sequence,divergent sequence and their examples and theorems
• Concept of monotonic sequence, recurrence relation, cauchy general principal of convergence
• Nested Interval theorem,Bolzano Weierstrass theorem,lim inferior and superior of the sequence,definition of series and theorems.
• The comparison test and its examples,Cauchy condensation test and its examples,convergence and divergence of series
• Alternating series,Leibniz series test,Root test,Ratio test,Dirichlet theorem and different example
• Mid Term Exams
• Introduction to limit of function,examples and sand-witching theorem
• Winter Break
• Concept of continuity and its examples,Intermediate Value Theorem
• Uniform continuity,theorems and examples,types of discontinuities and further applications Concept of differentiation in real analysis and its examples,
• The Chain Rule,Local maxima and minim and its geometric representation,Generalized Mean Value theorem
• Geometric representation of M.V.T, Lagrangian mean value theorem,Darboux’s theorem,L’Hospital rule and their examples ,Taylor’s theorem
• Final Term Exam