# Analysis I

 Analysis I, MAT320, 4 credits Section Monday Wednesday, 11:00-12:50 am, GI 305 Prerequisite: MAT 226 or equivalent vector calculus Text: Mathematical Analysis: a straightforward approach K.G. Binmore, 2nd Edition Cambridge University Press Professor C. Sormani Office Hours: Monday Wednesday 10-11 am, 3-4 pm Office: Gillet Hall 200B Email: sormanic (at) member.ams.org Webpage: http://comet.lehman.cuny.edu/sormani has a link to this course's webpage

Course Description: Introduction to Real Analysis; the real number system, limits, continuity, the mean value and Taylor's theorems and applications, Riemann integration and improper integrals.

Grading Policy: 11 Quizes: 5% each... Midterm Exam: 15%... Final:30%...

THE QUIZES COUNT MORE TOWARDS THE GRADE THAN EITHER OF THE EXAMS! Quizes will be in the first 10 minutes of class on Mondays. Students must submit completed *ed homework from the prior week in order to begin a quiz and that homework will be part of the score. Students who miss quizes because they are absent, late or have not completed the homework will receive a 0 on that quiz. There will be no make up quizes. There is one extra quiz.

Note that a quiz may contain proofs and the proofs should be written in a two column format with statements and justifications. Very similar proofs will be on the homework that is required to take the quiz. If you are unsure of the proof of a problem, you may write a question mark next to the step you are unsure of.

In the following syllabus, homework is written below the lesson when it is assigned and should be completed before the next meeting. All quizes and homework collection of *ed problems are on Mondays.

Syllabus: (will be updated regularly)

• Mon 8/27: Proofs and Counter examples
HW: Read 1.1-1.4, rewrite examples 1.5 and 1.6 as two column proofs, do 1.8/exercise *1,*2,*4,
• Wed 8/29: Quantifiers
HW: Complete the quantifiers worksheet, Read 1.7 and rewrite as a two column proof, do 1.8/ *5, *6, Read 7.1-7.5, Rules of Proof website
• Wed 9/5: Quiz 1 (must show 1.8/1,2,4,5,6 are all completed),
1.12/1*,2,3*,5,6; 1.20/2*, 3*, 6*; 7.16/2*,3*,4*
• Mon 9/10: Continuoum
• Mon 9/17: Quiz 2 and Sup and Inf
• Wed 9/19: Induction
HW: Read 3.7-3.9, Read handout, Do the four starred problems on the handout. Do 3.11/1i*,1ii*, 4*, (the handout is on my door)
Extra Credit due Mon: Do 1.20/4, 3.6/6,
Rules of Proof website
• Mon 9/24: Quiz 3 and Convergence
HW: Read 4.1-4.5, Do 4.6/1*,2*,3* (be sure to use epsilon in these)
• Wed 9/26: Sandwich
HW: Read 4.7-4.10, Do 4.20/3*, Do 3.11/2*, 3* Read 4.11-4.17, rewrite the proof in 4.17 and examples 4.18 as two column proofs, Read 4.19, Do 4.20/1*,2, 6*
• Mon 10/1: Quiz 4 and Monotone Sequences
HW: Redo quiz 3. Prove that a sequence which is increasing and bounded above converges to its sup.
Prove that if an converges to A and bn converges to B then (2an + 3 bn) converges to (2A +3B).
Read the rest of chapter 4, do 4.29/2,4.
• Wed 10/3: Subsequences (5.1-5.7)
HW: Redo all starred problems and quizes.
• Wed 10/10: Midterm Exam (on material from Quizes 1-4 and their homework)
HW: Go over homework assigned on 10/1, it will count as a starred HW for Quiz 5, Read 5.1-5.7, do 5.7/ 1*, 3*, This will also be on the quiz.
• Mon 10/15: Bolzano Weierstrass Theorem
HW: study for quiz 5 and do the above
• Wed 10/17: Quiz 5, liminf, limsup, Cauchy sequences
HW: Go over lesson notes and rewrite description about max/sup/limsup in terms of min, inf, liminf. Read 5.8-5.14, Do 5.15/ 1*, 3*, 4*, 5*, 6*, Use 6 to prove Bolzano Weierstrass. Read 5.16-5.19 Do 5.21/1*, 4*
• Mon 10/22: Quiz 6 and Limits of Functions
Read pages 70-72 of Larson, Do 28*, 29*, 34*, 37*,
HW: Read 8.1-8.5, Do 8.15/2*,3*, Prove Prop 8.12 (i) using Defn in 8.3,
• Wed 10/24: Continuity
read the rest of the handout from Larson and prove last theorem rigorously*,
Read 8.6-8.7, Do 8.15/ 6*, Read 8.8, Prove Prop 8.12 (ii) and (iii) using Theorem 8.8*, Read 8.13-8.14, Do 8.15/ 5*
• Mon 10/29: Quiz 7 and Continuity
HW: Read 8.6, 9.1-9.3 Prove 9.4(i)(ii)(iii) see hint below the three statements, Read 8.16, Prove 9.5* and 9.6*, Read 9.10, 9.13-9.14, Prove 9.10 imitating the proof in 9.9, Read 9.12 Prove 9.12 for infimum*, Do 9.17/1, 2*,
• Wed 10/31: Differentiation
HW: Read 9.16, Do 9.17/ 3*, 4*, 5*, 6* Read 10.1-10.3, 10.4-10.10, Do 10.11/2*, Read 10.12-10.14 Do 10.15/2*, 5*
• Mon 11/5: Quiz 8 and Mean Value Theorem
• Wed 11/7: Area and Integration
HW: Read 13.1-13.3 Write proofs for 13.4,13.5,13.6,13.7, Read 13.8-13.15, Study 13.11 and 13.15 closely. Study 11.2 closely.
• Mon 11/12: Quiz 9, Riemann Integral and Improper Integrals
• Wed 11/14: Lebesgue Integration
HW: Study 13 closely
• Mon 11/19: Quiz 10 on 13.4-13.7 and Series
HW: Read 6.1-6.11 and bring questions, Do: prove that the series 1/2 + 1/4 + 1/8 + 1/16 =1 using proof by induction to verify the partial sums add up to 1-(1/2^k) and using an epsilon N limit proof to show those partial sums converge to 1. Do this homework alone and hand it in on Monday.
• Mon 11/26: Convergence of Series
Review 4.4, 4.10, 4.17, 5.16, 5.17, 5.19, then read 6.7-6.19 carefully. You need to know these proofs very well. You may wish to look these tests up in your old calculus textbook and see the discussion there as well. Then, without consulting the text, write out rigorous proofs of 6.2, 6.5, 6.6, 6.13, 6.15, 6.17 and 6.18. Note 6.18 is tricky because it has limsup in the hypothesis rather than lim. Finally go back to 6.15 and read the last paragraph below the proof. Find the step in your proof which needs the stronger hypothesis.
• Wed 11/28: Conditional Convergence
HW: Catch up on Monday assignment, Read 6.20-6.25, Do 6.26/2*,3*,5* Study proofs of 6.13*, 6.15*, and 6.17* for the quiz.
• Mon 12/3: Quiz 11 and Convergence of Functions
Uniform and Pointwise convergence in Marsden Chapter 5.1 (see door for handouts) homework is Quiz 11: prove the convergence root test Thm 6.18 of Binmore
• Wed 12/5: Taylor Series and Convergence
HW: Read Marsden 5.6, Read Binmore 15.1-15.9, Read classnotes available on door, pick up Quiz 11 on door Do 15.6/ 1*, 2*, 6*
• Mon 12/10: Differentiating Taylor Series
HW: Read 15.7-8, Do 15.6/2* to hand in on Wed as a quiz
• Wed 12/12: Review for Final
The final will have two sections: short answers and proofs. In the proof section there will one proof: either a proof that a function is continuous at a point, or a sequence converges. In the short answer section other proving techniques will be tested including how to write the first line of a proof by contradiction, what the format of a proof by induction is, and how to fill in the justification for a given statement. In order to complete this section you will need to know the statements of all the important theorems and definitions we've learned this semester including sup, inf, bound, limit, bounded increasing sequences converge, sandwich lemma, subsequences of bounded sequences converge, Cauchy sequences, liminf, limsup, theorems about these, continuity, theorems about this, differentiation, mean value theorem, rolle's theorem, Riemann integration of continuous functions, theorems about integration, improper integrals, series, convergence tests including comparison, ratio, root and alternating series tests, Taylor series, radius of convergence, uniform convergence. You will also need working knowledge of these concepts in the sense that you must be able to find the limit, liminf, and limsup of various given sequences, the sup and inf of various sets and functions, the Taylor series for a given function and give its radius of convergence.
• Final: Wed Dec 19 11:00am-1:00 pm GI 305
• Due to the difficulty of the final, there will be a review session on Thursday Jan 10 10am-12pm in Gillet 205.
• Students may then retake the final on Monday Jan 28 10-12 am. This is the ordinary meeting time for Analysis II and Differential Geometry. At that time students who have not taken Analysis I with me will be given a review of proving techniques lesson. Another professor will proctor the exam.