Course Contents
The algebraic and order properties of set of real numbers. Supremum and infimum. Boundedness. Completeness of set of real numbers. Elementary topology of the real line. Extended real number system.
Sequences and their limits. Limits theorems. Monotone sequences. Subsequences. The convergent sequences and Cauchy’s sequences. The divergent sequences.
Infinite series. Comparison tests. Absolute convergence. Test for absolute convergence. Tests for nonabsolute convergence. Power series.
Limit of a functions. Continuous functions. Combination of continuous functions. Continuous functions on intervals. Uniform continuity. Monotone and inverse function. Continuity and compactness. Continuity and connectedness.
Differentiability. Chain rule. Inverse functions. The mean value theorem and its applications. Taylor’s Theorem and its applications. Relative extrema. Convex functions.
Course Learning Outcomes
Principle of Real Analysis can be well applied to the real world. It is used as a tool to handle machinery of mathematical language. The study of many important and advanced concepts becomes easy if the notation of sequences is employed.
Sets and Operations on Sets ( Lec-1 )
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Relation and Types of Relations(Lec-2)
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Types of Functions (Lec-3)
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Functions Continued...(Lec-4)
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Algebraic and Order Properties of Real Numbers (Lec-5)
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Methods of Proofs ( Lec-6)
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Methods of Proofs Continued ( Lec-7)
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The Completeness Property of Real Numbers ( Lec- 8)
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Complete and Incomplete Ordered Fields ( Lec-9)
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Properties of Suprimum and Infimum ( Lec-10)
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Properties of Suprimum and Infimum Continued (Lec -11)
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Maximum, Minimum, Maximal and Minimal Elements of Subsets of Real Numbers ( Lec -12)
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Archimedean and Condensation Properties ( Lec-13)
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Existence and Uniqueness of Positive Nth Root of Positive Numbers (Lec -14)
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Schwarz Inequality and Euclidean Spaces (Lec -15)
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Sequences and Their Convergence (Lec-16)
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Bounded Sequences and Limit Theorems (Lec-17)
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Applications of Sandwich Theorem (Lec-18)
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Monotone Bounded Sequences (Lec-19)
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Applications of Monotone Convergence Theorem (Lec-20)
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Monotone Subsequence and Bolzano Weirstrass Theorems (Lec-21)
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Cauchy Sequence (Lec-22)
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Contractive as Cauchy (or Convergent) Sequence of Real Numbers (Lec -23)
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Divergent and Properly Divergent Sequences (Lec-24)
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Infinite Series: A brief Introduction (Lecture -25)
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Convergence of Positive Term Series and Cauchy Criterion for Series (Lecture -26)
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Cauchy Condensation Test With Applications (Lecture -27)
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Comparison, Limit Comparison, Ratio and Root Tests (Real analysis Lec-28)
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Alternating Series Test. Dirichlet’s and Abel’s Tests (Lec-29)
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Grouping of Series (Lec-30)
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Rearrangement of Series (Lec-31)
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Limits of Functions (Lec-32)
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Limits of Functions Continued (Lec-33)
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Sequential Criterion for Limits of Functions (Lec-34)
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Divergence Criterion for Limits and Applications (Lec-35)
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Bounded Functions (Lec-36)
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Bounded Functions Continued...(Lecture-37)
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Bounded Functions and Compact Sets ( Lec-38)
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Continuity at a Point: Definition ( Lec-44)
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Proofs of Limit Theorems ( Lec-39)
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Limit Theorems With Applications ( Lec-40)
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Extensions of Limits: Definition and Examples (Lec-41)
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One Sided Limits for Monotone Function (Lec-42)
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Limits: Monotone Functions Continued (Lec-43)
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Classifications of Discontinuities (Lec-45)
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Book Title : Introduction to Real Analysis
Author : R.G. Bartle and D.R. Sherbert
Edition :
Publisher : John Willey & Sons
Book Title : Principles of Mathematical Analysis
Author : W. Rudin
Edition :
Publisher : McGraw-Hill
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