Lecture Notes

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====== Jad Damaj ======

About Me: I'm a first year math major from Nevada. I enjoy playing guitar and running.




{{  :math104-s22:s:jdamaj:104_notes.pdf | Course Notes}} 
===== Course Journal =====

==== Jan 18 ====

  * Overview of Course
  * Discussed natural numbers, integers, and rationals 
  * Problem with rationals: has holes which prevent us from getting sharp bounds on subsets
  * {{ :math104-s22:s:jdamaj:math_104_hw1.pdf | Exercises}}

==== Jan 20 ====

  * Rational Zeros Theorem 
  * Construction of $\R$ from $\Q$: Dedekind Cuts vs. Cauchy Sequences 
  * Completeness Axiom + Archimedean Property for $\R$
  * Definition of limits/convergence for sequences 

==== Jan 25 ====

  * Showed Convergence Sequences are bounded. 
  * Defined operations on convergent sequences. 
  * Showed some useful limits.

==== Jan 27 ====

  * Monotone Sequences 
  * Recursive Definition of Sequences 
  * lim inf and lim sup of a Sequence

==== Feb 1 ====

  * Cauchy sequences 
  * Cauchy sequences always converge in $\R$
  * Subseqeunces
  * Cantor's Diagonal trick to produce a convergent subsequence

==== Feb 3 ====

  * All sequences have a monotone subsequence 
  * All bounded sequences have a convergent (monotone) subsequence 
  * If $S$ is the set of subsequential limits of $s_n$, then sup$S$ = limsup$s_n$ and inf$S$ = liminf$s_n$

==== Feb 8 ====
  * limsup(a_nb_n) = lim(a_n)limsup(b_n) for convergent series $a_n$ with limit greater than 0
  * Introduced Series
  * "Sanity Check" and Comparison Test
  * Root and Ratio Tests

==== Feb 10 ====
  * Series
  * Summation by Parts
  * Power Series

==== 5 Questions ====
  * What is a good way to approach coming up with inequalities to use in proof, as in the Rudin exercises this week. 
  * What are some good counterintuitive counterexamples to keep in mind when working on problems. 
  * What specific properties of absolute convergence should we be familiar with for the exam, eg. rearrangements etc. 
  * What properties does multiplication in limsup(a_nb_n) have in general.
  * Is there a good way to get intuition for accumulation of infinite series, eg. the case of sum(1\n)

==== February 22 ====
  * Definition of Metric Space + examples
  * Topology 
  * Open Sets

==== February 24 ====
  * More Metric Space examples
  * Sequences + Cauchy Criterion 
  * Closure/ Closed Sets

 ==== March 1 ====
  * Continuous Maps (open cover def and sequential def)
  * Inherited Topology 

==== March 3 ====
  * Open cover compactness
  * Sequential compactness

==== March 8 ====
  * Sequential Compactness $\to$ Open Cover Compactness

==== March 10 ====
  * Connectedness

==== March 15 ====
  * Continuous maps preserve compactness and connectedness
  * Uniform Continuity
  * Discontinuity

==== March 17 ====
  * Sequences and Series of Functions
  * Uniform Convergence 

==== March 29 ====
  * Differentiation
  * Rolle's Theorem

==== March 31 ====
  * Generalized Mean Value Theorem
  * L'Hopital's rule

==== April 7 ====
  * Higher Derivatives
  * Taylor's Theorem

==== April 12 ====
  * Taylor Series
  * Power Series
  * Reimann Integral

==== April 14 ====
  * Integration
  * Reimann - Stieltjes Integral

==== April 19 ====
   *  Reimann - Stieltjes Integral

==== April 21 ====
  * Properties of Integrals

==== April 26 ====
  * Uniform Convergence with Integration
  * Uniform Convergence with Differentiation
===== Homework =====

  * {{ :math104-s22:s:jdamaj:math_104_hw1v2.pdf | Hw 1}}
  * {{ :math104-s22:s:jdamaj:math_104_hw2.pdf |Hw 2}}
  * {{ :math104-s22:s:jdamaj:math_104_hw3.pdf |Hw 3}}
  * {{ :math104-s22:s:jdamaj:math_104_hw4.pdf |Hw 4}}
  * {{ :math104-s22:s:jdamaj:math_104_hw5.pdf |Hw 5}}
  * {{ :math104-s22:s:jdamaj:math_104_hw6.pdf |Hw 6}}
  * {{ :math104-s22:s:jdamaj:math_104_hw7.pdf |Hw 7}}
  * {{ :math104-s22:s:jdamaj:math_104_hw8.pdf |Hw 8}}
  * {{ :math104-s22:s:jdamaj:math_104_hw9.pdf |Hw 9}}
  * {{ :math104-s22:s:jdamaj:math_104_hw10.pdf |Hw 10}}
  * {{ :math104-s22:s:jdamaj:math_104_hw11.pdf |Hw 11}}
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