# 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 $\infty$