**Course Information**

Official
stuff |
Stuff
specific to 131AH/1 Winter 2003 |

(*) - The course description and textbook and schedule are for 131A, not 131AH,
so these pages are only a very rough guide as to what to expect from the
course.

We recommend that you read the lecture notes **and** the
textbook concurrently with (or prior to) the lectures. If you only read these
sources occasionally and after the fact (e.g. when your homework is due) then
you will not get the most out of the course.

- Week 1: Introduction to analysis;
the natural number system; induction; the integers; the rationals
- Week 2: Cauchy sequences; the real
numbers; the least upper bound property; cardinality of sets
- Weeks 3/4: Countable and
uncountable sets; sup, inf, limsup and liminf; limits and limit points
- Week 5: Exponentiation, standard
limits, series, convergence tests, rearrangement of series
- Week 6: Subsequences;
Bolzano-Weierstrass theorem; limiting values of functions; continuity;
left and right limits
- Weeks 7/8: Maximum principle;
Intermediate value theorem; Uniform continuity; Differentiability; Mean
value theorem; Inverse function theorem
- Week 9: Partitions of intervals;
piecewise constant functions; the Riemann integral; Riemann sums;
properties of the Riemann integral
- Week 10: Riemann-Stieltjes
integrals; the fundamental theorems of calculus; products and absolute
values of Riemann integrable functions; integration by parts; change of
variables formula

**Sample exams and solutions**

- Solutions to HW1: page 1 page 2 page 3 page 4 page 5
- Solutions to HW2: page 1 page 2 page 3 page 4 page 5 page 6 page 7
- Solutions to HW3 and HW4
- Solutions to HW5
- Solutions to HW6
- Solutions to HW7
- Solutions to HW8
- Solutions to HW9
- Some stuff about the midterm
- Solutions to first midterm
- Solutions to second midterm

**Miscellaneous**

- A supplemental handout on logic.
You may also enjoy these lighter-hearted logic
puzzles of Lewis Carroll as practice.
- A supplemental handout on set
theory. (Errata: on line 21 of
page 8, “(d) Show that (A\B) U B = A
[...]” should be "(d) Show that (A\B) U B = A U B [...]". Thanks to Edoardo Buscicchio for this
correction.)
- A supplemental handout on the
decimal system (optional reading)
- The notion of two sets having equal cardinality if
there is a bijective mapping between them is intuitively obvious, but
there are still some subtleties if one tries to extend this idea beyond
just counting sets. Take a look at the missing
square puzzle or the Leprechaun puzzle to
see some of those subtleties.
- Real analysis was developed by several key
mathematicians, mostly in the nineteenth century, including Cantor,Cauchy,Dedekind,Hilbert,Lebesgue,Peano,Riemann,
and Weierstrass;
these links point to biographies of these famous mathematicians.