18. SHOW THAT A SEQUENCE OF REAL NUMBERS IS CONVERGENT IF AND ONLY IF IT IS CAUCHY || REAL ANALYSIS

TO WATCH ALL COURSES OF B.S MATHEMATICS 5TH SEMESTER CLICK THE LINK BELOW    • B.S MATH || 5TH SEMSETER|| PUNJAB UNIVERSI...   ------------------------------------------------------------------------------------------------------------------------------------------------------- TO WATCH COMPLETE PLAY LIST OF REAL ANALYSIS-I B.S MATHEMATICS 5TH SEMESTER CLICK THE LINK BELOW    • REAL ANALYSIS | B.S MATHEMATICS  | PUNJAB ...   ------------------------------------------------------------------------------------------------------------------------------------------------------- This course is specially designed for students of B.S MATHEMATICS 5th semester and students doing M.SC MATHEMATICS annually from Punjab university Lahore Pakistan. ------------------------------------------------------------------------------------------------------------------------------------------------------- COURSE: REAL ANALYSIS- I||PUNJAB UNIVERSITY|| 5TH SEMESTER|| CODE MATH-301 LEVEL: B.S MATHEMATICS CHAPTER # 2 SEQUENCE AND SERIES LECTURE # 18 In this lecture I shall discuss an important theorem which is known as CAUCHY'S GENERAL PRINCIPLE OF CONVERGENCE which can be stated as (1) Show that a sequence of real numbers is convergent if and only if it is Cauchy sequence. ------------------------------------------------------------------------------------------------------------------------------------------------------- FOLLOWING ARE THE CONTENTS OF REAL ANALYSIS- I Real Number System • Ordered sets, fields, the field of real numbers • Completeness property of R • The extended real number system • Euclidean spaces • Finite, countable and uncountable sets Sequences and Series • Sequences, subsequences, convergent sequences, Cauchy sequences • Monotone and bounded sequences, Bolzano Weierstrass theorem • Series, series of non-negative terms • Partial sums, the root and ratio tests, integral test, comparison test • Absolute and conditional convergence Limit and Continuity • The limit of a function • Continuous functions • Types of discontinuity • Uniform continuity • Monotone functions Differentiation • The derivative of a function • Mean value theorems, the continuity of derivatives • Taylor’s theorem Functions of Several Variables • Partial derivatives and differentiability, derivatives and differentials of composite functions • Change in the order of partial derivative, implicit functions, inverse functions, Jacobians • Maxima and minima ------------------------------------------------------------------------------------------------------------------------------------------------------- In this video, we'll explore Cauchy's general principle of convergence, which is a fundamental concept in analysis and is used to determine if a sequence of real or complex numbers converges. We'll start by discussing the definition of a Cauchy sequence and how it differs from a convergent sequence. Then, we'll delve into Cauchy's general principle of convergence, which states that a sequence of real or complex numbers converges if and only if it is a Cauchy sequence. We'll provide examples to help you better understand this principle, and we'll discuss its applications in various fields of mathematics. By the end of this video, you'll have a solid grasp of Cauchy's general principle of convergence and be able to apply it to your own mathematical problems. ----------------------------------------------------------------------------------------------------------------------------------------------- Share, Support, Subscribe!!! About: Maths 4 U is a You Tube Channel, where you can learn Mathematics in very simple way. In this channel you can find Mathematics Videos for Intermediate and B.S Mathematics. Further in my channel you can find videos for following Entrance Exams (1) ECAT (2) CSIR (3)GATE (4) NET (5) JEE MAIN (6) JEE ADVANCED (7) IIT-JEE (8) AIEEE, DCE (9) EAMCET, WB -JEE (10) ORISSA JEE & BIHAR CET (11) MP & RAJASTHAN PET (12) KERALA PET (13) PUNJAB & HARYANA CET & OTHER ENGG. INSTITUITES Here you can find Challenging Math's Problems related to daily life. Further you can find MATHS videos for the preparation of COMPETITIVE EXAMS like as CSS, PMS and PPSC. Here you can also find lectures on use of some Mathematics software's like as Math Type, LaTex, Matlab and Mathematica. #Maths4U. #professorazmatali #realanalysis #bernoulli_inequality