Contents
Preface xi
1. Real Numbers and Monotone Sequences 1
1.1 Introduction; Real numbers 1
1.2 Increasing sequences 3
1.3 Limit of an increasing sequence 4
1.4 Example: the number e 5
1.5 Example: the harmonic sum and Eulers number 8
1.6 Decreasing seq
1.1 Introduction. Real numbers.
Mathematical analysis depends on the properties of the set R of real numbers,
so we should begin by saying something about it.
There are two familiar ways to represent real numbers. Geometrically, they
may be pictured as th
Preface
This book is for a one-semester undergraduate real analysis course, taught
here at M.I.T. for about 25 years, and in its present form for about 15. It runs
in parallel with a more dicult course based on point-set topology.
Its origins Many years a
18.100A Introduction to Analysis Practice Final 3 hours
Directions. You can use the textbook, but no other material.
To justify the arguments, quote theorems, by name or number, verifying their hypotheses
where necessary.
Work problems totalling 100. You
18.100A Practice Problems for Exam 2 F-12 120 minutes (exam is 80 min.)
Directions: You can only use the book; cite relevant theorems when asked to.
1. A function f (x) has three distinct zeros a0 < a1 < a2 on an interval I , and in addition
f (a2 ) = 0.
18.100A Practice Questions for Exam 1
This could take 2 hours or more to complete and check over; the actual exam 1 will be
shorter and easier, about 2/3 the length. You can use the material in the book, Chs. 1-8,
but no other material (notes, problem set
18.100A Fall 2012: Assignment 26 (Optional, not for handing in)
This assignment gives exercises from chapter 23, for those studying the optional mate
rial of the Friday and Monday lectures. Solutions to these exercises will be sent by e-mail.
Reading: (Fr
18.100A Fall 2012: Assignment 25
As before, list collaborators, if any; its illegal to consult assignment solutions from previous semesters.
Reading: (Mon.) 27.1-.2; 27.3 (statement only) Improper integrals depending on a
parameter: uniform convergence, M
18.100A Fall 2012: Assignment 24
As before, list collaborators, if any; its illegal to consult assignment solutions from pre
vious semesters.
Reading: 26.2-.3; 21.3 Leibniz formula, Fubinis theorem. Gamma function.
1. (1) Work 26.2/2 .
2. (2) Work 26.2/5
18.100A Fall 2012: Assignment 23
The rules are the same as for previous assignments.
Reading Mon.: 25.1-.3 Constructing closed and open sets; characterization of compact
sets.
1. (2) Work 25.1/4ab, using and citing theorems in Chapter 25.
2. (2) Work 25.2
18.100A Fall 2012: Assignment 22
The rules are the same as for previous assignments.
Reading: 24.6-.7; p. 364; Notes on Open and Closed Sets (e-mailed)
Compact sets in R2 ; three theorems about a continuous function on a compact set S;
Open sets and close
18.100A Fall 2012: Assignment 21
The rules are the same as for previous assignments.
Reading Mon.: 24.1-24.5 The Euclidean and uniform distances (norms) on R2 .
Sequences in R2 , limits, continuous functions on R2 , Sequential Continuity theorem.
1. (1) I
18.100A Fall 2012: Assignment 20
The rules are the same as for previous assignments.
Reading: 22.5, 22.6 (through p. 319)
Dierentation of sequences and series; application to power series.
The end result of applying the theorems of this chapter to power s
18.100A Fall 2012: Assignment 19
Directions:
Same as for previous assignments.
Reading: 22.3,.4 Continuity of uniform limits. Integration of series term-by-term.
Problem 1. (2) Prove the series
enx
converges for x [0, ), and its sum f (x)
(x + n)2
n=1
is
18.100A Fall 2012: Assignment 18
The rules are the same as for previous assignments.
Reading: 22.1, 22.2
Uniform convergence for sequences and series of functions. The M-test for series.
1. (1) 22.1/1a
2. (2) Same as 22.1/1, for
3. (2) 22.1/3
4. (2) 22.2/
18.100A Fall 2012: Assignment 17
Directions: If you collaborate, list collaborators and write up the solutions independently.
Consulting solutions to problem sets of previous years is not allowed. Cite signicant theorems or examples when needed for reason
18.100A Fall 2012: Assignment 16
Directions: Same as before. This is longer (20 pts.) than usual, but the questions are not
dicult, and a lot of them deal with familiar calculus techniques.
For the two Questions, try to do them without looking at the answ
18.100A Fall 2012: Assignment 15
Directions You can collaborate, but should list collaborators, and write up the solutions
independently, i.e., thinking through the problem by yourself and expressing it in your own
words. Consulting solutions to problem s
18.100A Fall 2012: Assignment 14
Directions: You can collaborate, but must write up the solutions independently. Consulting solutions to problem sets of previous years is not allowed. Cite signicant theorems.
Reading: (Mon.) 15.1-.4 Dierentiation and loca
18.100A Fall 2012: Assignment 13
Directions: Same as for previous assignments. If you collaborate, list collaborators and
write up independently. No consulting previous semesters problem sets. Cite signicant
theorems.
Reading: 14.1-.3 Dierentiation: Local
18.100A Fall 2011: Assignment 12
Directions: You can collaborate, but must write up the solutions independently, and list
collaborators. Consulting solutions to problem sets of previous semesters is not allowed.
Cite signicant theorems being used in your
18.100A Fall 2012: Assignment 11
Directions: Same as before: write up solutions independently if you collaborate; list
collaborators, dont consult solutions from previous semesters; cite theorems or examples to
justify arguments which use them.
Reading: 1
18.100A Fall 2012: Assignment 10
Directions: Same as before you can collaborate, but write up solutions independently
and list collaborators; no consulting previous semesters solutions; cite theorems being used.
Reading: 11.4, 11.5
Discontinuity types, co
18.100A Fall 2012: Assignment 9
Directions: You can collaborate, but must write up the solutions independently, i.e.,
thinking through the problem by yourself and expressing it in your own words. Consulting
solutions to problem sets of previous semesters
18.100A Fall 2012: Assignment 8
Directions: Dont consult solutions to previous semesters assignments ; if you collaborate, write up the solutions independently and list collaborators.
Reading: Chaps. 9, 10
Chapter 9 should be review of things you have had
18.100A Fall 2012: Assignment 7
Directions: List collaborators; do not consult assignments from previous semesters; cite
relevant Theorems or Examples.
Reading: 8.1, 8.2 (omit Abel), 8.3; 8.4 (8.4: statements only, omit proofs)
Power series: radius of con
18.100A Fall 2012: Assignment 6
Directions: List collaborators, dont consult solutions to problem sets from previous semesters.
Reading: Mon.: 7.1-.2, 7.4-.5 Wed.: 7.3, 7.6-.7
Innite series; convergence tests; absolute and conditional convergence.
Problem
18.100A Fall 2012: Assignment 5
You can collaborate, but must list your collaborators, and write up the solutions independently. Cite by name or number any signicant theorems you are using in your arguments.
Consulting solutions to problem sets of previou
18.100A Fall 2012: Assignment 4
Directions: You can collaborate, but should list those you worked with and write up
the solutions independently (i.e., not copying but thinking them through by yourself).
Consulting solutions to problem sets of previous sem