Date

What we discussed/How we spent our time

Aug 26

Syllabus. Policies. Text.
Review of Math 2001.
We discussed how the axioms of set theory
allow the construction of the natural numbers,
$\mathbb N = \langle \{0,1,2,\ldots\}; 0, 1, +, \cdot\rangle$.

Aug 28

We discussed how to construct $\mathbb Z$ from $\mathbb N$,
$\mathbb Q$ from $\mathbb Z$, and we started discussing how to construct
$\mathbb R$ from $\mathbb Q$.

Aug 30

We defined finite sequences, infinite sequences,
the limit of a sequence of rational numbers,
Cauchy sequences of rational numbers,
null sequences. We defined a real number to be
an equivalence class of Cauchy sequences of rational
numbers modulo null sequences. We explained how to
define $0, 1, +, , \cdot, <$ on $\mathbb R$.
This handout
on ordered fields
was circulated.

Sep 4

We discussed the axioms for ordered fields,
and gave some examples and nonexamples.

Sep 6

We worked on this handout.
Then we discussed why any ordered field contains
an isomorphic copy of $\mathbb N$.

Sep 9

We introduced maximum elements, upper bounds,
least upper bounds, suprema, and the completeness axiom.
[The dual concepts are minimum elements, lower bounds,
greatest lower bounds, infima, and the completeness axiom.]
Quiz 1.

Sep 11

We discussed
a construction
of a nonarchimedean
ordered field, $\mathbb R(t)$, whose elements
are rational functions in the variable $t$
with coefficients in $\mathbb R$. The order is defined
by saying that $\frac{a_mt^m+\cdots+a_1t+a_0}{b_nt^n+\cdots+b_1t+b_0}$
is positive in $\mathbb R(t)$
iff $\frac{a_m}{b_n}$ is positive in $\mathbb R$.
We will see that the completeness property does not hold in
$\mathbb R(t)$.
We started investigating consequences of the Completeness Axiom
which hold in any complete ordered field,
but fail in $\mathbb R(t)$.
Today we showed that the Nested Interval Property holds in
any complete ordered field.

Sep 13

We showed that a complete ordered field $\mathbb F$ has the following
properties:
$\mathbb F$ is archimedean.
$\mathbb Q_{\mathbb F}$ is dense in $\mathbb F$.
$\mathbb F = \bigcup_{n=1}^{\infty} [n,n]$.
For every $r\in \mathbb F$ there exists $m\in\mathbb Z_{\mathbb F}$
such that $m1\leq r< m$.
We examined $\mathbb F = \mathbb Q$ and $\mathbb R(t)$ to see which
of these examples has the Nested Interval Property, the Archimedean Property,
or has a dense set of rationals.

Sep 16

We discussed this handout on
cardinality.
Quiz 2.

Sep 18

We continued to discuss cardinality.
We proved the CantorBernsteinSchroeder Theorem,
and explained why
$\mathbb N = \mathbb Z = \mathbb Q$.

Sep 20

We discussed ``Cantor diagonalization''.
We used it to prove (i) $[0_{\mathbb R},1_{\mathbb R}]$
is uncountable, (ii) ${\mathcal P}(\mathbb N)$ is uncountable,
and (iii) there is no surjective function from
a set $X$ to ${\mathcal P}(X)$ for any set $X$.
We explained why the following three sets have the same size:
(i) ${\mathcal P}(\mathbb N)$, (ii) the set of
(characteristic) functions $f:\mathbb N\to \{0,1\}$, and
(iii) the set of paths through a complete binary tree.
We proved that if $\mathbb F$ is an ordered field satisfying
the Nested Interval Property, then the cardinality of
its unit interval $[0_{\mathbb F}, 1_{\mathbb F}]$ is at least
${\mathcal P}(\mathbb N)$. (We did this by embedding the set
of paths through the complete binary tree into the set
$[0_{\mathbb F}, 1_{\mathbb F}]$.)
We stated
that if $\mathbb F$ is an ordered field satisfying
the Archimedean Property, then the cardinality of
its unit interval $(0_{\mathbb F}, 1_{\mathbb F})$ is at most
${\mathcal P}(\mathbb N)$. (Proof postponed until next lecture.)

Sep 23

We discussed the following results.
An ordered field is Archimedean iff its rationals are dense.
If $\mathbb F$ is an Archimedean ordered field
and $\mathbb K$ is a complete ordered field, then
there is a unique embedding of $\mathbb F$
into $\mathbb K$.
Any two complete ordered fields are isomorphic.
An ordered field is complete iff it is Archimedean and satisfies
the Nested Interval Property.
The cardinality of an ordered field
with the Nested Interval Property
is at least ${\mathcal P}(\mathbb N)$;
The cardinality of an Archimedean ordered field
is at most ${\mathcal P}(\mathbb N)$;
the cardinality of a complete ordered field
is exactly ${\mathcal P}(\mathbb N)$.
Quiz 3.

Sep 25

We discussed limits of sequences.
Most of the discussion was about how to read,
write, and understand formal sentences, like
$$(\forall \varepsilon>0)(\exists N)(\forall i)((i>N)\to (a_iL<\varepsilon)).$$
The main concepts/terms discussed were:
The alphabet of symbols appropriate for
writing about a structure (variables, logical symbols, nonlogical symbols, punctuation).
Defined symbols (we gave the formula for ``$y=x$'' that is valid for any ordered field).
Sentence formation rules.
Formula trees.
Restricted quantifiers.
Truth versus provability.
Quantifier games for determining truth
(with $\forall$belard and $\exists$loise).
In particular, we discussed how to describe
a winning strategy for $\exists$ or $\forall$.
Logic.
Practice!

Sep 27

We defined a metric, or distance function.
(Defining properties: a metric on a set $A$ is a 2variable function
$d: A\times A\to \mathbb R$ that is positive definite,
symmetric, and satisfies the triangle inequality.)
The metric used in $\mathbb R$ is $d(a,b) = ab$.
So the definition of limit,
$$(\forall \varepsilon>0)(\exists N)(\forall i)((i>N)\to (a_iL<\varepsilon)),$$
should be thought of in the following form:
$$(\forall \varepsilon>0)(\exists N)(\forall i)((i>N)\to (d(a_i,L)<\varepsilon)).$$
We gave a fake proof that limits are unique.
Then we gave a correct proof.
Finally, we proved that a convergent sequence is bounded.

Sep 30

We discussed the Algebraic Limit Theorem.
Quiz 4.

Oct 2

We discussed the Order Limit Theorem
and the Monotone Convergence Theorem.
Midterm Review!.

Oct 4

Practice
with the Monotone Convergence Theorem!

Oct 7

We defined subsequences, and discussed the theorem
that if $(a_n)_{n\in\mathbb N}$ converges to $L$,
then any subsequence also converges to $L$.
From this we developed a divergence criterion.
Quiz 5.

Oct 9

Midterm Review!.

Oct 11

Midterm!

Oct 14

We reviewed the meaning and importance of
``definitions'' and
``winning strategies''.
We proved the BolzanoWeierstrass Theorem.
No Quiz!

Oct 16

We recalled the definition of Cauchy sequence.
We worked on a handout
to prove that any convergent sequence is bounded.
We began a discussion of why any
Cauchy sequence of real numbers is convergent.

Oct 18

We finished the proof of the Cauchy Criterion.
We discussed
different kinds of limiting processes,
in particular
infinite series.

Oct 21

We discussed the definition of ``$\sum a_i = L$''.
We discussed the following facts about series.
(Sometimes with only sketches of proofs.)
Cauchy Criterion for series. (with proof)
Comparison Test. (sketchy proof)
Series of positive terms converge iff their partial sums are bounded.
(with proof)
$N$th term test for divergence. (with proof)
Deleting first $N$ terms does not affect convergence.
(sketchy proof)
The Harmonic Series diverges. (with proof)
The sum of the even terms of the Harmonic series divereges,
as does the sum of the odd terms. (sketchy proof)
Quiz 6.

Oct 23

We proved (or reviewed the proofs of):
the Comparison Test.
Deleting first $N$ terms does not affect convergence.
Hence if two terms are eventually equal, they both converge
or they both diverge.
that a series of nonnegative terms converges if and only if
its partial sums are bounded.
the Geometric Series Theorem.
the Absolute Convergence Test.
the Limit of a Telescoping Series Theorem.
As an example of the last item we explained why $\sum \frac{1}{n(n+1)} = 1$.

Oct 25

We discussed results about series that have both
positive and negative terms, including:
the Absolute Convergence Test.
the Alternating Series Test.
Dirichlet's Test. (This handout is relevant.)
Dirichlet's Theorem asserting that
if an absolutely convergent series converges to $L$,
then any series obtained from it by rearrangement also
converges to $L$.
the Riemann Rearrangement Theorem.
As an example for Dirichlet's Test, we explained why
$\sum \frac{\cos(k)}{k}$ converges.

Oct 28

Snow day!

Oct 30

Snow day!

Nov 1

We began a discussion of the topology of the real line.
Topology glossary!
Quiz 7.

Nov 4

We continued discussing the topology glossary,
focusing on metric spaces and the metric topology.
We defined the $\ell_r$ metrics on $\mathbb R^2$,
and drew pictures of the unit balls under
$\ell_1, \ell_2, \ell_{\infty}$.
Quiz 8.

Nov 6

We reviewed the Riemann Rearrangement Theorem,
and explained how to rearrange the terms of the alternating
harmonic series to obtain a series that converges
to any prescribed number in $[\infty,\infty]$,
or to nothing at all.
We discussed the sequence of generalizations
from $\mathbb R$, to an arbitrary metric space,
to an arbitrary topological space.
We discussed how one determines whether
a set is open in a metric space
compared to how one determines whether
a set is open in a topological space.
We gave examples of sets that are open but not closed,
closed but not open, both closed and open, and
neither open nor closed. We ended the lecture with
a definition of limit point, and a definition
of closed set in terms of limit points.

Nov 8

We proved that a subset of a metric space is closed iff
it contains all of its limit points.
We gave some examples to illustrate this theorem.
We defined the Cantor set, and explained why it
is a closed set of cardinality
${\mathcal P}(\mathbb N) \;\;(= {\mathbb R})$.

Nov 11

We discussed this handout,
especially the part about compactness.
We defined cover, subcover, boundedness,
and explained why any compact subset of
of a metric space is closed and bounded.
We also explained why any closed and bounded
subset $A$ of a metric space has the property
that any sequence in $A$ has a subsequence that converges
to a point of $A$.
Quiz 9.

Nov 13

We proved the HeineBorel Theorem.

Nov 15

We defined connectedness and continuity.
We argued that a subset of $\mathbb R$
that is connected must be an interval.

Nov 18

We proved that intervals are connected.
We stated Bolzano's Theorems, and derived
the Extreme Value Theorem from one of them.
Quiz 10.

Nov 20

Proofs of Bolzano's Theorems.

Nov 22

Continuity at a point.
Dirichlet's function and Thomae's function.
Algebraic Continuity Theorem.

Dec 2

Definition of uniform continuity.
Proof of the HeineCantor Theorem, which asserts that
a continuous function on a compact set is uniformly continuous.
We also explained why a continuous periodic function (like $\sin(x)$)
is uniformly continuous.
nonQuiz worksheet!.

Dec 4

We discussed the three types of discontinuities:
removable, jump, and essential (or oscillation) discontinuities.
We defined the derivative, and
discussed the Algebraic Differentiability Theorem
and the the theorem asserting that differentiable functions are continuous.
Finally, we discussed the results of Darboux:
Theorem. If $f$ is differentiable
on $[a,b]$ and $f'(a)< C < f'(b)$, then there
is some $c\in (a,b)$ such that $f'(c)=C$.
Corollary. If $f$ is differentiable
on $[a,b]$ and $f'$ is not continuous at $c\in (a,b)$,
then the discontinuity of $f$ at $x=c$ is essential.
Review for Final Exam!.

Dec 6

We drew pictures of differentiable functions
with discontinuous derivatives.
We proved the Interior Extremum Theorem. We proved
Darboux's Theorem (if $f$ is differentiable
on $[a,b]$, then $f'$ has the IVP on $[a,b]$).
We proved that corollary to Darboux's Theorem
(if $f$ is differentiable
on $[a,b]$ and $f'$ is not continuous at $c\in (a,b)$,
then the discontinuity of $f$ at $x=c$ is essential).
Worksheet! (A continuous
nowheredifferentiable function.)

Dec 9

No Quiz!
We worked on the Blancmange worksheet.
Solutions!
We discussed pointwise and
uniform limits of sequences and series
of functions (Definitions 6.2.1, 6.2.1B, 6.2.3),
The Uniform Limit Theorem (Theorem 6.2.6),
and the Weierstrass Mtest (Theorem 6.4.5).
We used this material to deduce
that the Blancmange function
is continuous everywhere.
The rest of class time was used in outlining
why the Blancmange function is differentiable nowhere.

Dec 11

We
reviewed for the final exam.
