Proving All the Sequence Limit Laws | Real Analysis

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Real Analysis course:    • Real Analysis  
Real Analysis exercises:    • Real Analysis Exercises  

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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:    • Proof: Triangle Inequality Theorem | Real ...  
Proof of Reverse Triangle Inequality:    • Proof: Reverse Triangle Inequality Theorem...  
Definition of the Limit of a Sequence:    • Definition of the Limit of a Sequence | Re...  
Proof that a Constant Sequence Converges to its Constant Value:    • Proof: Constant Sequence Converges to its ...  
What are Bounded Sequences?    • What are Bounded Sequences? | Real Analysis  
Proof that a Convergent Sequence is Bounded:    • Proof: Convergent Sequence is Bounded | Re...  

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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