Chapter 1

Real Number System and Completeness

Algebraic and order properties of R

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Bounded and unbounded sets

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Supremum and infimum

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Completeness property of R

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Archimedean property

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Chapter 2

Sequences of Real Numbers

Definition and convergence of sequences

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Limit theorems for sequences

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Monotone sequences

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Cauchy sequences

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Bolzano-Weierstrass theorem

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Chapter 3

Infinite Series and Convergence Tests

Convergence of infinite series

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Comparison test

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Ratio and root tests

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Alternating series

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Absolute and conditional convergence

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Chapter 4

Limit and Continuity of Functions

Epsilon-delta definition for functions

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Sequential criterion for limits

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Algebra of limits

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Continuous functions on intervals

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Properties of continuous functions on closed intervals

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Chapter 5

Uniform Continuity and Differentiability

Uniform continuity definition

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Non-uniform continuity examples

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Continuous extensions

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Differentiability definitions

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Caratheodory's theorem

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