Math 104: Introduction to Real Analysis (2021 Fall)

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Instructor: Peng Zhou
Email: pzhou.math@berkeley.edu
Office: Evans 931
Office Hour: Monday 12:10-1pm, updated Wednesday 10:10-11am, Friday 10:10-11am

Lecture: MWF, 11:10am - 12:00. Etcheverry 3107.

GSI: James Dix. Mondays 9am-6pm, Wednesday 12-2pm in Evans 1049.

Online Help:

• Zoom chat channel: search for “Math 104 with Peng Zhou”, then you will find the channel. I will answer question there.
• My zoom office: https://berkeley.zoom.us/j/97935304012 time by appointment.

• Elementary Analysis: The Theory of Calculus, by Kenneth A. Ross. springer link (UC login required).
• Principles of Mathematical Analysis, by Walter Rudin
• Introduction to analysis, by Terry Tao. ( springer link )
• notes from 2021 spring previous version

20% homework; 2 midterms 20% + 20%; and final 40%. If you didn't do well in one of the midterm, you have the option to drop it, and final will have a 60% weight. The lowest homework grades will be dropped.

Midterm date: 9/22 (Wed), 11/3 11/10 (Wed). There will be no make-up midterms.

Final Date: Mon, Dec 13 • 11:30A - 2:30P

Homework will be submitted via gradescope. Entry Code:YVZRDZ

part 1: number system, sequence and limit, series.
part 2: metric space and topology. continuity.
part 3: differentiation and integration.

• Aug 25: introduction. counter-examples. Tao, Ch 1.
• Aug 27: Peano Axioms for natural numbers. (Tao Ch2). Inadequacy of $\Q$. Least upper bound (Rudin Ch 1, section 1)
• Reading homework; Tao Ch 1 and 2. Rudin Ch 1.
• HW 1 (with solution): due next Tuesday (Aug 31) 6pm

• Aug 30: More about sup. Definition of field.
• Sep 1: Cauchy sequence of rational numbers.(Tao 5.2)
• Sep 3: Equivalent Cauchy sequences of rational numbers as real numbers. Operation and properties of real numbers. (Tao 5.3)
• HW 2 (with Solution): Due next Thursday 6pm. (Due date changed)

Tao 5.3-5.5

• Sep 8: arithmetic operation on $\R$.
• Sep 10: order on $\R$, and least upper bound property of $\R$.
• HW 3 with solution: due next Tuesday (Sep 14) 6pm

Tao Ch 6. Ross Ch 2.1 - 2.7.

• Sep 13: Sequences in $\R$. Convergent implies Cauchy. Arithmetic operation commute with limit. Bounded monotone sequences are convergent. limsup.
• Sep 15: $\pm \infty$. Tao 6.4. Cauchy sequences are convergent.
• Sep 17: Finish Cauchy sequence is convergent. Limit points and subsequence.
• HW 4 Due next Tuesday 6pm
• Midterm 1: Review

• Sep 20 Subsequences, Countable set, $\R$ is not countable.
• Sep 22 Midterm 1
• Sep 24 Various results from Ross section 10-12. (No office hour today)
• HW 5 due next Thursday 6pm.

• Sep 27 Ross section 12
• Sep 29 Series Ross 14,15. Root and Ratio test.
• Oct 1 finishing series, integral test. Start Metric space and topology.
• HW 6, Due next Thursday 6pm. (All future homeworks will be due on Thu 6pm)

• Oct 4 Open sets in metric spaces.
• Oct 6 Examples of Metric spaces and topology. Metric on Graph. Metrics on $\R^2$, $l^1, l^2, l^p, l^\infty$ metric.
• Oct 8 Limit points and closure.
• HW 7, Due next Thursday 6pm

• Oct 11 Closure and Interior. Open covers and Compact sets
• Oct 13 Compact sets are closed. Closed subset of compact set is compact. Compactness is absolute notion. (Rudin 2.30, 2.33, 2.34, 2.35)
• Oct 15 Towards Thm 2.41. Finishing compactness. (will not talk about perfect set). sequential compactness and compactness
• HW 8 Due next Thursday 6pm.

Wrapping up Ch 2. Continuity. Rudin Ch 4. Another concise lecture note to follow is Rui Wang's lecture note https://math.berkeley.edu/~ruiwang/pdf/104.pdf

updated office hour from now on Tuesday 11-12am moved to Wednesday 10:10-11am

• Oct 18: Wrapping up loose ends in Ch 2: connected set. Sequential compactness and compactness. More examples.
• Oct 20: Begin Rudin Ch 4. Two definitions of continuous functions, using $\epsilon-\delta$, and use open sets.
• Oct 22: Example of Continuous functions. Do pre-image and image of continuous functions preserve open / closed / bounded / compact sets?
• HW 9 Due next Thursday 6pm

Continuity.

• Oct 25: Connectedness and Continuity.
• Oct 27: Operations on continuous function. (Cartesian product, composition, restriction of domain and codomain)
• Oct 29: Limit of a function and discontinuity.
• HW 10: Due next Thursday 6pm.

Midterm 2 postponed to next Wednesday.

• Nov 1: Monotonic Functions (Rudin p95-98)
• Nov 3: Uniform Continuity (Rudin p90-91), Intermediate Value Thm (Rudin Thm 4.23)
• Nov 5: Review for Midterm 2
• HW 11 Due next Friday 6pm.

Sequences of functions. (Rudin Ch 7)

Office hour on Monday moved to Tuesday 12-2pm

• Nov 8: Pointwise convergence and Uniform Convergence.
• Nov 10: Midterm 2 policy about cheat sheet is the same as midterm 1
• Nov 12: Uniform convergence preserves continuity. Examples.
• No HW this week

Rudin Ch 5, Differentiation. One can also see notes from 2021 spring previous version

• Nov 15: definition. examples. Chain rule.
• Nov 17: mean value theorem.
• Nov 19: L'hopital rule. Smooth Functions. Taylor theorem.
• HW 13. Due Nov 29 Monday.

Rudin Ch6

• Nov 22 A brief encounter with Lebesbue measure theory and integration (optional). The definition of Riemann integrable functions.
• Nov 24 No class.
• Nov 26 No class

• Office hour of GSI changed this week: 3pm - 6pm Tuesday and 9am-4pm Wednesday.
• Videos from past semester are available on bcourse media gallery. You can use them for review.

• Nov 29: Continuous function and Monotone functions are Riemann integrable.
• Dec 1: Riemann Stieltjes integral
• Dec 3: Fundamental Theorem of Calculus
• HW 15, this is only for practice, not due.

Review week. No class. We have daily office hours 12-1pm, at zoom link, from Monday-Thursday. If you plan to come, please arrive by 12:10.

• Tuesday office hour will be held in-person only. 12:10noon-1pm.

final exam and solution, Common Mistakes in Final, final-grades