This course covers the fundamentals of sequences, including limits, convergence, and various limit examples. Students will learn about bounded sequences, infinite limits, Cauchy sequences, and the construction of real numbers. The teaching method includes theoretical explanations, limit examples, proofs, and application-based questions. This course is intended for individuals interested in advanced mathematics and those preparing for entrance exams in mathematics.
Overview
Syllabus
What is a sequence?.
Limit of a Sequence.
The Basics (Limit Example 1).
Simple Fraction (Limit Example 2).
DON'T factor out! (Limit Example 3: Complex Fraction).
The Limit Does NOT Exist (Limit Example 4).
Limits are unique.
Square Root Limit (Limit Example 5).
Bounded Away from Zero.
Sum of limits.
Convergent sequences are bounded.
Absolute Value (Limit Example 6).
Squeeze Theorem Proof.
Product of Limits.
Reciprocals (Limit Example 7).
Exponentials (Limit Example 8).
DON'T use logarithms (Limit Example 9).
Infinite Limits (Limit Example 10).
Infinite Limit Laws.
Limit Duality Theorem.
Power vs Exponential limit (Limit Example 11).
Monotone Sequence Theorem.
Golden limit.
Babylonian Square Root.
Monotone Sequence implies Least Upper Bound.
Decimal Expansions.
What is Limsup ?.
Limsup vs Limits.
Limsup Squeeze Theorem.
Cauchy Sequences.
Completeness.
The Legend of Z.
Real numbers Cauchy construction.
What is a Subsequence?.
Inductive Construction.
Another Inductive Construction.
Monotone subsequence.
Can limsup be attained?.
Limit Points.
The Bolzano Weierstraß Theorem.
Direct Bolzano Weierstraß.
Can you solve this UC Berkeley math PhD entrance exam question?.
A neat diagonal argument.
Limsup Product Rule.
Limsup Sum Rule.
Pre-Ratio Test.
Limit with n!.
Not your average YouTube video.
Taught by
Dr Peyam
