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This note is outcome of Fall 24 Math 444. Elementary Real Analysis in UIUC, taught by professor Marius Junge.
The number after sub-heading indicates the number of class(e.g. Def 37.3 means 3rd point in lecture #37) and the book is Introduction to Real Analysis 4th Edition by Robert G. Bartle.

Topics

Key points for sequences:

• Cauchy sequence
• Cs for sets: complete, closed, compact, connected
• sequential version: sequentially Closed, sequentially compact
• C for functions: continue, convergent
• sequential version: sequentially continue
• useful tools: Lipschitz, bounded and totally bounded, Delta-ball, Metric Space

Key points for integrable:

• integrable, integral
• Darboux integral and Riemann integral
• differentiable
• mesh and partition

Some proofs:

• $\pi >=2>=2\sqrt{2}$ --- Proofs of pi
• Set $S$ is totally bounded $⟹$ $S$ is bounded --- 3 lemmas of class 28
• Set $S$ is compact $⟹$ $S$ is bounded set --- proof of Thm 21.7
• For Metric Space $X(C,d)$, $C$ is closed $⟹$ $X$ is complete --- proof of Thm 15.1
• Riemann integral of ${\lim }_{a⟶1}\sum cos(a^{\frac{j}{2}})a^j(1-a)$ --- Final Review problems