Thomas Dahlke 

Notes are organized by topic collected together with both Ross and Rudin.


====== Thomas Dahlke ======

===== Links to resources and rages I have found helpful:  =====


Youtube Channels:
Wrath of Math: Large Real Analysis Playlist that goes over many definitions and proof explanations.
 [[https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli|Link to W.o.M. youtube playlist]]

Dr. Peyam (Used to teach math at UC Berkeley!): Extremely large bank of Real and Complex Analysis. Videos are broken up by topic where he goes into more explanation and examples for each subject. 

Topics with sub-playlists include:
  - Real Numbers
  - Sequences 
  - Topology 
  - Series 
  - Limits and Continuity

[[https://www.youtube.com/c/DrPeyam/playlists|Link to Dr. Peyam's Youtube playlists ]]


Other Misc. Links :

====== Midterm #1 ======

Ross Chapter 1
  - §1 The set N of Natural Numbers
  - §2 The set Q of Rational Numbers
  - §3 The Set R of Real Numbers
  - §4 The completeness Axiom 
Link to Textbook Notes:
{{ math104-s21:s:midterm_1_ch_1.pdf |}}

Ross Chapter 2
  - §7 Limits of Sequences
  - §9 Limit Therorems for Sequences
  - §10 Monotone Sequences and Cauchy Sequences 
  - §11 Subsequences 
  - §12 lim sup's and lim inf's
  - Exrta: Intro to Proofs
Link to Textbook Notes:
{{ math104-s21:s:midterm_1_ch_2.pdf |}}

====== Midterm #2 ======

Rudin Ch.2, Ross §13

Rudin Ch.2 Basic Topology 
  - Finite, Countable, and Uncountable Sets
  - Metric Spaces
  - Compact Sets
  - Perfect Sets
  - Connected Sets

Ross Ch 2. Sequences 
§13 *Some Topological Concepts in Metric Spaces

Link to Textbook Notes: {{ math104-s21:s:rudin_ch_2_ross_13.pdf |}}


Ross Ch 2. Sequences
  - §14 Series
  - §15 Alternating Series and Integral Tests

Link to Textbook Notes: {{ math104-s21:s:ross_14_15.pdf |}}

Rudin Ch. 4 Continuity
  - Limits of Functions 
  - Continuous Functions 
  - Continuity and Compactness
  - Continuity and Connectedness 
  - Discontinuities 
  - Montotonic Functions 
  - Infinite Limits and Limits at Infinity 

Link to Textbook Notes: {{ math104-s21:s:rudin_ch_4.pdf |}}

Ross Ch. 4 §24, §25, §26; Rudin Ch. 7 (sections 1-2)

Ross Ch. 4 - Sequences and Series of Functions 
  - §24 Uniform Convergence
  - §25 More on Uniform Convergence
  - §26 Differentiation and Integration of Power Series

Rudin Ch. 7 - Sequences and Series of Functions 
  - Uniform Convergence
  - Uniform Convergence and Continuity 

Link to Textbook Notes: {{ math104-s21:s:ross_24_25_26_rudin_7.pdf |}}


====== Midterm 2 - Final  ======

Rudin Ch. 5 - Differentiation 
  - The Derivative of Real Functions
  - Mean Value Theorems 
  - The Continuity of Derivatives
  - L'Hospital's Rule 
  - Derivatives of Higher Order
  - Taylor's Theorem 

{{ math104-s21:s:rudin_chapter_5.pdf |}}


Rudin Ch. 6 - The Riemann-Stieltjes Integral 
   - Definition and Existence of the Integral 
   - Properties of the Integral 
   - Integration and Differentiation

{{ math104-s21:s:rudin_chapter_6.pdf |}}


Rudin Ch. 7 - Sequences and Series of Functions 7.16-7.18 
  - Uniform Convergence and Continuity
  - Uniform Convergence and Integration

{{ math104-s21:s:rudin_chapter_7.pdf |}}