# Analysis I

 Analysis I, MAT320, 4 credits Section Monday Wednesday, 11:00-12:50 am, 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: Mon/Wed 9:45-11:00 am, Wed 2:00-3:30pm 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! 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 *ed problems are collected on the days of quizes.

Syllabus: (will be updated regularly)

• Mon 8/31: Quantifiers
• Wed 9/2: 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, Rules of Proof website
• Wed 9/9: Proof Workshop
HW: Read 1.7 and rewrite as a two column proof, do 1.8/ *5, *6, Read 1.9-1.20,
1.12/1*,2,3*,5,6; 1.20/2*, 3*, 6*; 7.16/2*,3*,4*
• Mon 9/14: Quiz 1 (must show 1.8/1,2,4,5,6 are all completed), Continuoum
• Wed 9/16: Sup and Inf, Archimedian Property
• Mon 9/21: Proof by 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 9/29: Do 1.20/4, 3.6/6,
Rules of Proof website
• Wed 9/23: Quiz 2 (on 1.20, 2.10, 3.6) and Convergence
HW: Read 4.1-4.5, Do 4.6/1*,2*,3* (be sure to use epsilon in these)
• No class Mon 9/28, meet Tuesday instead:
• Tue 9/29: Sandwich Lemma
HW: Do 3.11/2*, 3*, Read 4.7-4.9 and go over class notes carefully,
Prove that if an converges to A and bn converges to B then (2an + 3 bn) converges to (2A +3B).
• Wed 9/30 : Quiz 3 (on Induction) and Monotone Sequences
HW: Read 4.14-4.16, Rewrite the proof in 4.17 and examples 4.18 as two column proofs, Read 4.19, Do 4.20/1*,2, 6* Prove that a sequence which is increasing and bounded above converges to its sup.
Read the rest of chapter 4, do 4.29/2,4.
• Mon 10/5: Quiz 4 (on Convergence) and Subsequences (5.1-5.7)
HW: start redoing all starred problems and quizes.
• Wed 10/7: Bolzano Weierstrass Theorem, Extra Credit Quiz on limits
HW: study for Midterm Exam (which will essentially be problems similar to those in Quizes 1-4)
• No class Columbus Day Mon 10/12
• Wed 10/14: Midterm Exam
HW: Quiz V will be about describing sequences using the definitions of bounded, increasing, decreasing, Thm 4.17, Thm 4.10, Thm 4.25 and Thm 5.2 as we did in class on 10/7. To prepare for this quiz, Read 5.1-5.3, go over lesson 10/7, Read 5.8-5.10 and examine what subsequence of a_n and b_n from 10/7 is monotone in the proofs there, then read 5.11-5.14, do 5.15/4,5,6.
• Mon 10/19: Quiz 5 (as described above) and Cauchy sequences
HW: ( Read 5.16-5.19 on Cauchy sequences, Do 5.21/1*, 4*, Quiz 5 Solutions
• Wed 10/21: 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,
• Mon 10/26: Quiz 6 (is just to hand in starred HW) and 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*
• Wed 10/28: 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*,
• Mon 11/2: Quiz 7 (group classwork due Wednesday)
Group classwork is: Prove 9.5 and 9.6, Prove 9.12 for infimum, Read 9.16, Do 9.17/2,3,4,5,6.
• Wed 11/4: Differentiation
• Mon 11/9: Quiz 8 (Continuity) and Mean Value Theorem
• Wed 11/11: Review of Calculus I from an advanced perspective
HW: Study all work on continuity and differentiation,
For Quiz 9: prove limit of sin(x)/x is 1 and limit of (cos(x)-1)/x is 0 using trigonometry and areas of triangles as in a calculus textbook but adding justifications from Analysis, then prove the derivative of sine is cosine and the derivative of cosine is sine.
• Mon 11/16: (Hand in Quiz 9) Area and Integration
Read the "Riemann Sum" article on wikipedia. Write the left sum, right sum, upper sum and lower sums for the following five integrals using evenly spaced intervals so that xi-xi-1=(b-a)/n. Find the relationship between epsilon and delta for each of these uniformly continuous functions as well and estimate how large n must be to guarantee an error of epsilon in each integral. Verify this works. Verify that the upper sum is larger than the lower sum and the left and right sums are in between, and that the upper sum minus the lower sum has an error less than epsilon when n is chosen large enough.
1) f(x)=4x integrated from 0 to 3
2) f(x)=5x+2 integrated from 3 to 8
3) f(x)=10-2x integrated from 1 to 4
4) f(x)= (x-2)2 + 1 integrated from 1 to 3
Prove the integral of a constant function f(x)=c from a to b is c(b-a), by taking the sums and explicitly evaluating them.
All of this is part of Quiz 10 due 11/23 (group work is encouraged)
• Wed 11/18: Riemann Integral
Read 13.1-13.3, Write proofs for 13.4 and 13.6, Read 13.16-13.17, Show that a function which is 0 on irrational numbers and 1 on rational numbers is not Riemann Integrable,
Look over 13.9-13.15, Read 13.19-13.22 carefully, look over 13.23-13.25, Do 13.26/1, 2.
All this homework plus the work assigned Monday counts as Quiz 10 (due 11/23) (group work is encouraged)
• Mon 11/23: (Hand in Quiz 10) Limits, Improper Integrals and Logs
All this homework counts as part of Quiz 11 (due Mon 11/30) (group work is encouraged)
• Wed 11/25: Series
HW: Review 4.4, 4.10, 4.17, then
Read 6.1-6.3, write out proofs of 6.2, 6.3
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.
This homework and Monday's homework is part of Quiz 11 (due Mon 11/30) (group work is encouraged)
• Mon 11/30: (hand in Quiz 11) Convergence of Series
Read 6.10-6.11, Write out a proof of 6.11
Review Cauchy sequences in 5.16, 5.17, 5.19, then Read 6.12-6.15
Rewrite the proofs of 6.16-6.19 (in 6.18 you may assume the limit is 0 rather than the limsup if that is easier for you).
• Wed 12/2: Taylor Series and Convergence of Functions
Homework: 1) find the Taylor series for e(3x) and check where it converges using the ratio test, 2) find the Taylor series for 1/(1-x) and check where it converges using the ratio test. 3) Find the Taylor series for Ln(x+1) and check where it converges. This will prepare you for the quiz on Monday.
• Mon 12/7: Quiz 12 (in class on Taylor Series), Differentiating Taylor Series
HW: Read 15.7-8, Do 15.6/2* to hand in on Wed as a quiz
• Wed 12/9: Review for Final
The final will have two sections: short answers and proofs. In the proof section there will be two proofs:
• a proof that a function is continuous at a point
• a proof that a sequence converges.
In the short answer section, other proving techniques will be tested including
• write the first line of a proof by contradiction,
• outline a proof by induction
• suggest theorems that might be relevant to prove a statement
• fill in the justifications for a proof about integrals
• find limits, sups and infs without proof
• find taylor series and verify convergence of the series
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, 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 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: Wednesday December 16 11am-1pm