MT321 Analysis I
  Fall 2014, 225 Higgins

Dan Chambers   365 Carney

chambers@bc.edu    617-552-3769 (email is more reliable)


Office Hours: Mondays 1-2, Wednesdays 3-4, Fridays 10-11 and by appointment.


We will cover the real number system (briefly mentioning complex numbers), countable and uncountable sets, metric spaces, open/closed sets, compact sets, connected sets, sequences and subsequences, limit points, numerical series, convergence theorems, continuous/discontinuous functions, continuity and compactness/connectedness, Cauchy sequences, the derivative, mean value theorems, continuity of derivatives, and interesting examples throughout the course.


The text we'll  use is Principles of Mathematical Analysis, third edition, by Walter Rudin.


Here is a link to the syllabus.


This is a work in progress; check back for updates and assignments.








Date
Section/Topic Covered 
 HW assigned
HW due
September


3
rational numbers
HW#1

5
ordered sets, sup and inf


8
ordered fields, real numbers, Archimedean property


HW#1
10 the rationals are dense in the reals, examples of sup and inf,

HW#2

12 Nested Interval Theorem, finite/infinite sets

15
countable/uncountable


17
conclusion, start of metric spaces, examples
HW#3
HW#2
19
limit points, closed sets


22 open sets, closure of a set begun



24
closure completed, compact sets begun

HW#3
26
more on compactness
HW#4

29
k-cells are compact


October



1
Heine-Borel theorem, connected sets


3
sequences, convergence
HW#4
6
norm of a linear space, properties of sequences

8
more properties



10 midterm 1  solutions


15
subsequences, Cauchy sequences


17
complete metric spaces
HW#5

20
monotone sequences, series

22
geometric series, Cauchy Condensation Test


24
ratio test

HW#5
27
root test
HW#6

29
integral test, alternating series


31
alternating harmonic series, rearrangements


November



3
rearrangements finished, functions started
HW#7
HW#6
5
limits of functions, sequential equivalence


7
continuity and compactness


10
continuity and connectedness, uniform continuity

HW#7
12
discontinuities
HW#8

14
continuity/discontinuity examples


17
the derivative


19
properties of the derivative

HW#8
21
midterm 2   solutions


24
examples


December


1
extrema, mean value theorem
HW#9

3
more on the MVT


5
intermediate value property


8
Cantor set

HW#9
10
Cantor set concluded, questions


18
Final 12:30 p.m.