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 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 $m-1\leq r< m$. 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 Cantor-Bernstein-Schroeder 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 $$(\forall \varepsilon>0)(\exists N)(\forall i)((i>N)\to (|a_i-L|<\varepsilon)).$$ 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 2-variable 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 $d(a,b) = |a-b|$. So the definition of limit, $$(\forall \varepsilon>0)(\exists N)(\forall i)((i>N)\to (|a_i-L|<\varepsilon)),$$ 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 Bolzano-Weierstrass 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 Heine-Borel 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 Heine-Cantor 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. non-Quiz 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 nowhere-differentiable 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 M-test (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.