I am looking for a book that covers introduction to real analysis. Currently, I am reading The Elements of Real Analysis, by Robert Bartle. However, I quickly noticed that about half of the theorems and all of the sample questions don't have solutions to them so it's hard for me to know if my answers are correct so I looks around and was able to find the following book on the internet Principles of Mathematical Analysis which does provide a solution manual.
When working through Rudin, even though you have a solutions manual, you should not give up on problems before you have solved them. There are problems in that book that take some of the best students hours over days to solve. The process of banging your head against the wall (or the book, or any other hard object) is part of the book and part of your preparation for mathematics. When you do get through Rudin, you will be in a very good place to step into the field of analysis $-$ possibly even the next Rudin book, Real and Complex Analysis.
bartle introduction to real analysis solutions manual
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As a soft introduction to analysis before Rudin, I would recommend my teacher's book: Mikusinski's An Introduction to Analysis: From Number to Integral. It is fairly short and easy to get through, and will prime your brain for the more intense fare of Rudin's book. It only covers single-variable analysis, however, which is 8 out of 11 chapters in Rudin; most courses in analysis only necessarily cover the first 7 chapters anyways.
"This text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. This third edition includes corrections as well as some additional material."
Analysis, one of the pillars (along with algebra and topology) of modern mathematics, begins with a rigorous development of single-variable calculus. Thus,this course serves the important purpose of teaching you how to rigorouslyprove and apply results in calculus, including results related to the notionsof limit, continuity, derivative, integral, and infinite series. Inherentin all of these notions is the concept of approximation. As we shallsee, a good grasp of this latter concept is essential not only in proving''pure'' results in analysis, but is also crucial in ''applied'' problemsrequiring estimations. In any approximation a key question is ``howdo you estimate the error''? In the first part of this course,we will look at some types of algebraic manipulations that can be usedin error estimation; we will also look at more powerful methods involvingthe mean value theorem for derivatives.Difference between the courses:MAT 320 is more comprehensive and providesa firm grounding for further study. MAT 319 has more of an emphasis ontopics which arise in high-school calculus.Students planning to go on to graduate schoolin mathematics are advised to take MAT 322 and MAT 324 as well.Students wanting to take MAT 322 or MAT 324 (or the seminarsMAT 401 or MAT 402) will need to take MAT 320, not MAT 319. Studentswho want to take these courses after MAT 319 instead will need to do some extrawork, and get permission from the relevant instructor.Prerequisites: C or higher in MAT 200 or permission of instructor;plus one of the following: C or higher in MAT 203, 205, 211, AMS 261, or A- orhigher in MAT 127, 132, 142 or AMS 161. Anyone lacking these prerequisitesrisks deregistration.Instructors: Daryl Geller, 4-100B Math Building,phone 632-8327. E-mail daryl@math.sunysb.edu - Website darylOffice hours: Tuesdays and Thursdays from 12:50-2:20. Check for announcements or postings on the website regularly!Michael Movshev, 4-109 Math Building, phone 632-8271.E-mail mmovshev@math.sunysb.edu - Website mmovshevOffice hours: to be announced. Teaching Assistants: Jan GuttE-mail jgutt@math.sunysb.eduXiaojie Wang, 2-106 MathE-mail wang@math.sunysb.eduOffice hours: Mondays and Wednesdays, 2-3 p.m. in the Math LearningCenter, and Mondays 1-2 p.m. in 2-106 Math.Text for the first five weeks and for MAT 320:D. Geller, A Bridge to Analysis, available through lulu.com.You can click on this link to order the book:A Bridge to Analysis Please select Ground shipping, as Mail may take too long.lulu.com is a self-publishing website that makes books available verycheaply.Solutions will be distributed as the semester progresses. There are solutions for almost every problem in the book.Text for MAT 319, after the first five weeks:R.G. Bartle and D.R. Sherbert, Introduction to Real Analysis, 3rd edition,available in the university bookstore. You can buy it after the split.Grading System: The first midterm, on the first three chapters of thebook (up to page 117) will be given in class on Thursday, October 7.It will be graded by Monday, October 11, at which time the classes will split.You will be allowed to switch your registration from MAT 319 to MAT 320 at that time.Your results on the test will help you to make an informed decision.There will also be a second midterm. In MAT 320, it will be held on Tuesday, November 16. The final examination will be held on Thursday, December 16, from11:15 a.m. - 1:45 p.m.Students are expected to ensure when they register for the courses thatthey will be available for the final examination, and that they donot have too many final exams on that date.The final course grades will be determined as follows:homework 10%, two midterms 25% each, final exam 40%.The grades of A- and A will be reserved for students who demonstrate asubstantial ability to apply the concepts of the course in new andsomewhat creative ways.Please note that there will be no curve in this course indetermining grades. Incompletes will be granted only if documented circumstances beyond yourcontrol prevent you from completing the course work.Homework:The only way to learn the material is to work problems for yourself. Eachweek, you should attempt to do all of the problems from the sectionswhich are covered in class. We will ask you to hand some problems in.Your homework will be graded meticulously and will give you vitalfeedback on where you are making mistakes.Homework is a means to an end, the ``end'' being for you tolearn the material. We encourage you to work on homework togetherwith friends. In this course, we will never prosecute anyone foracademic dishonesty on any issue relating to homework.If you hand in complete, correct solutions, you will get fullcredit for them, no matter how you obtained them. When you hand in homework in this course, you are not claiming that it is your own work.If someone regularly ``does''the homework by copying from friends or from solution manuals, they areonly cheating themselves, since this is not a way to learn the material.Moreover, they will not receive the benefits of the feedback that our meticulous grading will provide.Never be shy to ask us how to do a homework problem, even if you handed in a copied solution that you do not understand. You will not beprosecuted or condemned for this, and we will be only too glad to helpyou. Approximate Course Schedule:Chapter 1: Aug 31 -- Sep. 14>Chapter 2: Sep. 16 -- Sep. 28Chapter 3: Sep. 30 - Oct. 5First Test (on Chapters 1-2): Oct. 7`After the split, MAT 320 continues:Chapter 4: Oct. 12 - Oct. 28Chapter 5: Nov. 2 - Nov. 11Second Test (on Chapter 4): Nov. 16Chapter 6: Nov. 18 - Dec. 9After the split, MAT 319 continues:To be announced. Americans with Disabilities Act: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room128, (631) 632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.Arrangements should be made early in the semester(before the first exam) so that your needs can be accommodated.
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