Math 521- Introduction to Real Analysis (Summer 2025) 2024/08/17 | No Comments Lec. 1 – Naïve set theory Lec. 1 – part B Lec. 2 – Existence of Roots Lec. 3 – Metric Topolgy p. I Lec. 3 – part B Lec. 4 – Metric Topolgy pt. II Lec. 5 – Cauchy sequences Lec. 6 – Compactness Lec. 7 – Numerical Sequences Lec. 8 – Numerical Series Lec. 9 – Sequences series continued Lec. 10 – Numerical Series pt. III Lec. 11 – Product of Series Lec. 12 – Metric Spaces and Cont. Functions Lec. 12-1 Alternating Series Lec. 12-2 Thin Sequence Theorem Lec. 12-3 products of series Lec. 13 – Product Theorem Proof Lec. 14 – Abstract Metric Spaces & Continuity I Lec. 14 – precise definition of a metric space Lec. 15 – Metric Spaces Lec. 15-2-Continuity Lec. 15-3 Theorem, TFAE nesting Lec. 16 – Metrix Spaces Continued Lec. 16-2-Intermediate value theorem Lec. 17-1- IVT Sequences of functions Lec. 17-2-Ordinary differential equations Lec. 17-3-part b Lec. 17-4-Notes1 Lec. 17-5-part c Lec. 17-6-Cauchy like criterion Lec. 17-7-Theorem def Continuity with bounded Lec. 18 – proof of completeness Lec. 18-2-Examples and key points Lec. 18-3-Uniform continuance Lec. 19 – The Riemann-Darboux Integral pt. IV Lec. 19-a-Rigorous calculus Lec. 19-b-Rules for differentiation Lec. 19-c-Rolle’S theorem Lec. 19-d-Mean value theorem Lec. 19-e-Higher derivatives & Taylor’s Theorem Lec. 20 – Bernstein approximation Theorem Lec. 20-a-derivatives of trig func. Lec. 20-b-Corollaries Lec. 20-c-The existence of pi Lec. 21 – Riemann Darboux Integral Lec. 21-a Proposition and Defs Lec. 21-b proof, lower sum leq upper sum Lec. 21-c Fundamental definitions Lec. 22 – examples of Integrable functions Lec. 22-a concrete examples Lec. 22-b-questions Lec. 22-c- example 2 Lec. 22-d- example 3 Lec. 22-e-useful remarks Lec. 22-f- Riemann Integral definition Lec. 23 – Two kinds of integrals Lec. 23-a- Main Theorem (easy direction) Lec. 23-b-TIE Lec. 23-c (hard direction) key lemma Lec. 23-d- definitions & justifying Lec. 24 – Examples and Exercises Lec. 24-a-properties of the Riemann Darboux integral Lec. 24-b- lipschitz Lec. 24-c- Definitions, characteristics indicator function Lec. 25- left over exercises Lec. 25-a-first half of Lebegues character Lec. 25-b-claims Lec. 25-c-Theorem H Lebesque Lec. 25-d-core lemma Lec. 25-e- proof of proportion Lec. 25-f-Finale College Notes, Daily Learning, Math, Real Analysis
