Date

What we discussed/How we spent our time

Aug 27

Syllabus. Policies. Text.
Review of Math 2001.

Aug 29

Axioms of set theory.
We discussed axioms (1)(7). We also
defined the natural numbers as the least inductive set.

Aug 31

We discussed how to read, write and interpret formal sentences,
such as $\forall x\exists y(x=y)$. We practiced on the
problems from the August 27 handout.

Sep 5

We discussed this
handout about functions. Then we discussed
how to construct $\mathbb N\to \mathbb Z\to \mathbb Q$.

Sep 7

We discussed how to extend addition
from $\mathbb N$ to $\mathbb Z$, and also
how to pretend that $\mathbb N\subseteq \mathbb Z$.
Finally we began the construction $\mathbb Q\to \mathbb R$.
We started by constructing the
rational sequences of small variation (``clustering sequences''),
which we called ``Cauchy sequences''. Then we defined
two Cauchy sequences to be equivalent if
interlacing them produces a Cauchy sequence.

Sep 10

We defined Cauchy sequences of rational numbers,
null sets, and the relation $\sim$
of ``equivalence modulo null sets''. The set of real numbers
is defined to be the set of Cauchy sequences of rational numbers
modulo null sets. We defined $+, , 0, \cdot, 1$, positive, and $\leq$
on $\mathbb R$.
Solution to Quiz 0.

Sep 12

We showed that addition of real numbers is well defined.
We showed that (using the fact that $\mathbb Q$ satisfies
$\forall x\forall y(x+y=y+x)$)
$\mathbb R$ also satisfies
$\forall x\forall y(x+y=y+x)$.
Finally, we went over the definition of
ordered field.

Sep 14

I circulated a handout defining ordered fields.
We finished the discussion of how the real numbers
can be constructed as the completion of the rationals
via Cauchy sequences. We also discussed the fact
that the completion of the reals via Cauchy sequences
is just another copy of the reals. We discussed
that the reals numbers, as constructed, are the only
ordered field up to isomorphism that satisfies the Completeness Axiom.

Sep 17

We characterized sup's (Lemma 1.3.8).
We explained why the Completeness Axiom implies its dual.
We proved that the intersection of a
nested sequence of closed intervals is nonempty (Theorem 1.4.1).
We ended class with a quiz.
Solution to Quiz 1.

Sep 19

We showed that any complete order field is
Archimedean.
We described an nonArchimedean ordered field.
We gave most of the proof that the rational
numbers are dense in any Archimedean ordered field.

Sep 21

We reviewed cardinality.
The main definitions were:
$A\leq B$,
$A=B$,
$A < B$, equipotent,
ordinal number, cardinal number, countable versus uncountable.
The main theorems were: Cantor's Theorem,
CantorBernsteinSchroeder Theorem,
The theorem asserting $A\times A=A$ for infinite $A$,
and the theorem asserting that a countable union of countable
sets is countable.
We recalled that
$$\mathbb N=\mathbb Z=\mathbb Q < {\mathcal P}(\mathbb N)=\mathbb R=\mathbb C.$$
We mentioned that the Continuum Hypothesis, $\mathbb R=\aleph_1$,
which is neither provable nor refutable from the axioms of set theory.

Sep 24

We defined infinite sequence, finite sequence,
convergence of a sequence and limit of a sequence.
We showed that a constant sequence $(c,c,\ldots)$
converges to $c$. We also showed that the
sequence $(0,1,0,1,\ldots)$ does not converge.
Solution to Quiz 2.

Sep 26

We worked on this handout.
Then we gave a fake proof that limits are unique.

Sep 28

We gave the correct proof that limits are unique.
Then we proved that
convergent sequences are bounded, and that
limits preserve some algebraic operations.

Oct 1

We proved that limits respect order.
We proved the Monotone Convergence Theorem.
Solution to Quiz 3.

Oct 3

We proved that a subsequence of a convergent sequence
converges to the same limit. We used this
to develop a divergence criterion, which applies
to show that $(0,1,0,1,0,1,\ldots)$ diverges.
We proved that $1 = .\overline{9}$, and that
\[
\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{\cdots}}}}=2.
\]

Oct 5

We proved the BolzanoWeierstrass Theorem and the Cauchy Criterion.

Oct 8

We defined infinite series, partial sums, and
convergence of infinite series.
We proved that if $\Sigma a_i$ is convergent, then $\lim a_i = 0$.
Solution to Quiz 4.

Oct 10

Review for the midterm!
Too many practice problems!

Oct 12

Midterm!

Oct 15

Convergence of series!
No quiz today!

Oct 17

Review of formulas.
The main point of the review was: don't mix up
syntax with semantics.
Then we discussed absolute convergence
versus conditional convergence. The main facts were these:
Any absolutely convergent series converges.
Rearranging the terms of an absolutely convergent series
does not change the limit.
(Riemann Rearrangement Theorem.)
By rearranging the terms of a conditionally convergent series,
one can produce a series whose limit is any value
in $[\infty,\infty]$.
A series $\sum a_n$ can be reordered to
be conditionally convergent iff
(i) $\lim a_n = 0$, and (ii) the series of its positive terms
diverges and the series of its negative terms diverges.
A power series converges absolutely within its radius of convergence.

Oct 19

Topology of the real line.

Oct 22

We reviewed some topology terminology, and
examined weird and normal examples of topologies.
(Trivial topology, cofinite topology, discrete topology,
a metric topology on ${\mathcal P}(Z)$ for finite $Z$,
and the metric topologies on $\mathbb R$ and $\mathbb R^2$.
Solution to Quiz 5.

Oct 24

We defined a set $K$ to be compact if every
sequence from $K$ has a convergent subsequence,
and every such subsequence converges to an element of $K$.
We proved that a subset
of the real line is compact iff it is closed and bounded.

Oct 26

We proved the HeineBorel Theorem.

Oct 29

We proved that the connected subsets of $\mathbb R$
are the intervals.
Solution to Quiz 6.

Oct 31

We introduced continuity, saw some examples
of discontinuous functions, proved the
Algebraic Continuity Theorem, and derived
that every ratio of polynomials is continuous
wherever the denominator is nonzero.

Nov 2

We proved that
continuous images of compact sets
are compact.

Nov 5

We proved the HeineCantor Theorem: a continuous function on a compact
set is uniformly continuous.
Solution to Quiz 7.

Nov 7

We proved that the continuous image of a connected set is connected.
We derived the Extreme Value Theorem and the Intermediate Value Theorem
from this.

Nov 9

Derivatives!
We defined the derivative. We proved that differentiability
implies continuity. We proved parts of the Algebraic
Differentiability Theorem. We saw an example of a
continuous function that is not differentiable, and
an example of a function differentiable at exactly one point.

Nov 12

We proved the Interior Extremum Theorem and Darboux's
Theorem, and saw an example of a differentiable function
with a discontinuous derivative.
Solution to Quiz 8.

Nov 14

We discussed Rolle's Theorem, the MVT
and the fact that if $g'=0$
on an interval then $g$ must be constant.
Then we began discussing an example of
a continuous, nowhere differentiable function.

Nov 16

We worked to prove the nowhere differentiability
of the blancmange function.
(Some hints for the handout.)

Nov 26

We discussed pointwise convergence and
uniform convergence of sequences and series
of functions. We showed that the pointwise
limit of a sequence of continuous functions need
not be continuous.
Solution to Quiz 9.

Nov 28

We discussed this handout.
We proved the Uniform Limit Theorem and the
theorem of the Weierstrass $M$Test.

Nov 30

Integration! We defined
partitions, upper and lower sums, the Riemann
integral and gave an example of a bounded
function that is not Riemann integrable.
(Overall, we covered everything
in Chapter 7 up to Theorem 7.2.9, together with
Example 7.3.3.)

Dec 3

We proved that continuous functions on
bounded intervals are integrable (Theorems 7.2.8, 7.2.9).
Solution to Quiz 10.

Dec 5

We proved Theorem 7.4.2(ii)(v) (properties of the integral)
and stated Theorem 7.4.4 (Integrable Limit Theorem).
We saw an example where $\lim\int f\neq \int\lim f$.

Dec 7

We discussed this handout,
then proved that the Integrable Limit Theorem.
We saw how to evaluate $\int_0^1 (1+x/n)^n dx$.
We discussed the statement of Dini's Theorem.

Dec 10

Measure zero. Oscillation. Lebesgue's Theorem
characterizing Riemann integrable functions.

Dec 12

Review for the final!
Practice problems!

Dec 18

Final exam, 4:307pm.
