======Real Analysis Handwritten Notes by Kaushef Salamat======
{{ :notes:real-analysis-kaushef-salamat.jpg?nolink&600|Real Analysis Handwritten Notes by Kaushef Salamat}}

We are very thankful to Ms. [[people:kaushef]] 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. [[:people: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====
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<col sm="6">
  * 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  
    </col>
<col sm="6"> 
  * 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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</grid>
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====Notes of Analysis====
{{topic>analysis&nouser&simplelist}}
{{tag>MSc BS Analysis Notes Kaushef_Salamat}}