19. CH # 2 SEQUENC AND SERIES || NESTED INTERVAL THEOREM || NESTED SEQUENCE || REAL ANALYSIS-I, GATE

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 # 19 In this lecture I shall discuss the following (1) Concept of Nested Sequences (2) Nested Interval Theorem which can be stated as Prove that the intersection of nested sequence of closed interval contain only one point ------------------------------------------------------------------------------------------------------------------------------------------------------- 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 ------------------------------------------------------------------------------------------------------------------------------------------------------- The Nested Interval Theorem is a fundamental theorem in mathematics that deals with the concept of nested intervals. It states that if you have a sequence of closed intervals, each contained within the next one, and the length of each interval approaches zero, then there exists a unique point that lies in all of the intervals. This theorem has many applications in various fields of mathematics, such as real analysis, topology, and number theory. It is used to prove important results such as the Bolzano-Weierstrass Theorem, which states that every bounded sequence of real numbers has a convergent subsequence. In this video, we will explore the Nested Interval Theorem in detail, discussing its history, statement, and proof. We will also look at some examples of how it is used in mathematics and discuss its significance in the field. Whether you are a math enthusiast or a student looking to deepen your understanding of mathematical concepts, this video is sure to be an informative and engaging exploration of the Nested Interval Theorem. ----------------------------------------------------------------------------------------------------------------------------------------------- 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