MATH 242: Analysis 1 (McGill Final)

Currently Playing: Proving All the Sequence Limit Laws | Real Analysis

Support the production of this course by joining Wrath of Math to access all my real analysis videos plus the lecture notes at the premium tier! https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join 🛍 Check out my math clothing store: https://mathshion.com/ Real Analysis course: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Real Analysis exercises: https://www.youtube.com/playlist?list=PLztBpqftvzxXAN05Gm3iNmpz9SkVfLNqC Get the textbook! https://amzn.to/3CMdgjI We'll prove the five basic limit laws for convergent sequences. The limit laws for the sum of convergent sequences, difference of convergent sequences, constant multiples of convergent sequences, product of convergent sequences, and the quotient of convergent sequences. There are, of course, other limit laws not proven here, but these are typically presented as "the sequence limit laws" or "algebraic limit theorems". They are our first batch of general laws for manipulating convergent sequences and their limits. Let a_n converge to a, b_n converge to b, and let c be a real number. The limit laws we prove are as follows... a_n+b_n = a+b a_n - b_n = a-b c*a_n = c*a a_n*b_n = a*b a_n/b_n = a/b provided b isn't 0 and each b_n isn't 0 Proof of Triangle Inequality Theorem: https://www.youtube.com/watch?v=051WWg9twO0 Proof of Reverse Triangle Inequality: https://www.youtube.com/watch?v=4OmIRB5Zv-4 Definition of the Limit of a Sequence: https://www.youtube.com/watch?v=cTnlHZD5ss4 Proof that a Constant Sequence Converges to its Constant Value: https://www.youtube.com/watch?v=O-quXJi6-ws What are Bounded Sequences? https://www.youtube.com/watch?v=hkCKQdAFLAw Proof that a Convergent Sequence is Bounded: https://www.youtube.com/watch?v=7N-4XZlxBho Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: https://crayonangel.bandcamp.com/ Follow Wrath of Math on... ● Instagram: https://www.instagram.com/wrathofmathedu ● Facebook: https://www.facebook.com/WrathofMath ● Twitter: https://twitter.com/wrathofmathedu TABLE OF CONTENTS 0:00 Motivating Example and Intro 3:37 Sum of Sequences 10:39 Difference of Sequences 17:07 Constant Multiple of a Sequence 22:18 Product of Sequences 36:01 Quotient of Sequences 51:52 Using the Limit Laws to Make a Nasty Example Easy as Pie