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## Lecture Topics

 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 non-Archimedean 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, Cantor-Bernstein-Schroeder 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 Bolzano-Weierstrass 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 Heine-Borel 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 Heine-Cantor 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, 4:30-7pm.