Real Analysis Handwritten Notes by Kaushef Salamat

Real Analysis Handwritten Notes by Kaushef Salamat

We are very thankful to Ms. Kaushef Salamat for providing these notes. Real Analysis is a core subject in BS or MSc Mathematics. Without a fundamental grasp of Real Analysis, one cannot claim to be a mathematician. These notes are very comprehensive containing almost all the notions of Real Analysis. For providing these notes, Ms. Iqra Liaqat has our sincere gratitude.

Name Real Analysis: Handwritten Notes
Author Kaushef Salamat
Pages 344 pages
Format PDF (see Software section for PDF Reader)
Size 8.52 MB

Contents & Summary

  • Introduce Sets
  • Methods of Proof
  • Contradiction
  • Contrapositive
  • Terminology
  • Ordered Set
  • Ordered Field
  • Bounded and Unbounded Sets
  • Archimedian Principle
  • Condensation Property
  • The Extended Real Number System
  • Absolute Value of a Real Number
  • Schwarz Inequality
  • Euclidean Space
  • Inner Product
  • Norm
  • Dedekind's Property
  • Inclusion Function
  • Inverse Function
  • Metric
  • Open Cover
  • Compact Set
  • Open Cover
  • Separable Sets
  • Disconnected
  • Sequences
  • Monotone Sequences
  • Euler Number
  • Subsequences
  • Urysohn Property
  • Monotone Subsequence Theorem
  • The Bolzano-weirstrass Theorem
  • Cauchy Sequence
  • Contractive Sequence
  • Properly Divergent Sequence
  • Infinite Limits
  • Oscillate Sequence
  • Properly Divergent
  • Comparison Theorem
  • Limit Inferior and Limit Superior
  • Cluster Point
  • Cauchy's Second theorem on Limit
  • Sets of Real Numbers
  • Heine-Borel (Covering) Theorem
  • Infinite Series
  • Cauchy Criterion
  • Consequence of Cauchy Criterion
  • Comparison Test
  • Limit Comparison Test
  • Absolute and Conditionally Convergent Series
  • Rearrangement of Series
  • Test for Absolute Convergence, Cauchy Root Test
  • Ratio Test, Raabe's Test, Bertrand's Test
  • Guass's Test, First Log Test, Second Log Test
  • Alternating Series
  • Abel's Lemma
  • Dirichlet Test
  • Abel's test
  • Limits (Limits of Functions)
  • Limits of Function at a Real Number
  • Sequential Criterion for Limits
  • Divergence Criteria
  • Bounded Functions
  • Sequeeze (Sandwich) Theorem
  • Some Extensions of the Limit Concepts
  • Monotone Function
  • Continuous Functions
  • Composition of Continuous Functions
  • Properties of Continuous Function
  • Extreme Value Theorem
  • Bolzano's Intermediate Value Theorem
  • Preservation of Intervals Theorem
  • Brouwer's Fixed Point Theorem
  • Continuous Inverse Theorem
  • Uniform Continuity
  • Continuous Extension Theorem
  • Pecewise Linear Function
  • Differentiation
  • Chain Rule
  • Inverse Function
  • Darboux's Theorem
  • Criterion for Integrability
  • Improper Integrals
  • Beta Function
  • Absolute Convergence
  • Infinite Range of Integration
  • Comparison Test for Convergence at $\infty$

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