https://courses.pzhou.org/ 2026-04-17T11:03:19+00:00 https://courses.pzhou.org/ https://courses.pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture-1&rev=1771684872&do=diff Lecture 1 Exercises: * Ross 1.10, 1.12 * Read Ross 'Rational Zero Theorems', then do 2.1, 2.2, 2.7 * Try proving Ross Theorem 3.1, 3.2, by yourself, without read his proof. It is a good exercise for logical deduction. Yes, the result may sounds obvious for $\Q$, but you need to prove them for text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_2&rev=1771684872&do=diff Lecture 2 Last time we recalled what is $\N$, $\Z$ and $\Q$. I hope you had realized that I did not cover much in class, and you do need to read the textbook (Ross) to learn the details. In particular, you want to work on exercises on induction, read and apply the rational zero theorem (to show that $\sqrt{2}$ is not a rational number). We also mentioned the word 'ring' and 'field', which are algebraic concepts, which you can read in Harrison's note. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_3&rev=1771684872&do=diff Lecture 3 First, we will get familiar with a few examples of convergent and non-convergent sequences. Then, we prove a few properties on convergent sequence, first, they are bounded; second, they can only converge to a single value. Then, we will prove some operations (+,-,/, etc) on convergent sequence. Finally, we prove a few useful examples of limit (Thm 9.7). See also text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_4&rev=1771684872&do=diff Lecture 4 This time we consider some abstract results as what sequences converges. * monotone bounded sequence converges. * what is $\limsup$ and $\liminf$? The $\epsilon$-of-room trick. * Cauchy sequence. Discussion time: Ex 10.1, 10.7, 10.8 in Ross text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_5&rev=1771684872&do=diff Lecture 5 * Cauchy sequence: “we don't know where we are heading, but we agree with each other more and more”. * Cauchy sequence is equivalent to convergent sequences.... in $\R$. * Limit sets and subsequences. Cauchy Sequences First, the definition. Let $(a_n)$ be a sequence in $\R$, we say $(a_n)$ is Cauchy, if for any $\epsilon> 0 $, we have $N>0$, such that for all $n,m>N$, we have $|a_n - a_m | < \epsilon$. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_6&rev=1771684872&do=diff Lecture 6 Last time, we ended at discussion of two equivalent definitions of subsequential limit. I hope the Cantor's diagonalization trick was fun. Today, we prove the following results * If $s_n$ converge to $s$, then every subsequence of $s_n$ converges to $s$. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_7&rev=1771684872&do=diff Lecture 7 * more about liminf and limsup. Ross 12 * Series. basic definition and examples. partial sum. Cauchy conditions. absolute convergence and convergence. root test, ratio test, alternating series test. Discussion: * Ross Ex 12.10, 12.12, 12.13 text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_8&rev=1771684872&do=diff Lecture 8 More about series following Rudin * Thm 3.39: radius of convergence for power series * Summation by part.Thm 3.41, Thm 3.42, Thm 3.44 * Rearrangement theorem of Riemann Discussion: * Rudin's exercises in Ch 3, 6,7,9,11 text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_11&rev=1771684872&do=diff Lecture 11 Exam solution given in lecture. We begin discussing metric space and topological space. The note from last year might be useful. Can you think of other example of metric spaces? text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_12&rev=1771684872&do=diff Lecture 12 We plan to discuss topological space. Read Ross section 13 to see the axioms of 'topology'. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_13&rev=1771684872&do=diff Lecture 13: Continuous function. Last time we had some basic notion of metric space (a set with the notion of distances), and topological space (which is a set with some notion of which subsets are open). Today, we will define continuous functions (or rather continuous maps) between metric spaces and between topological spaces. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_14&rev=1771684872&do=diff Lecture 14: Compactness There are two notions of compactness, they turn out to be equivalent for metric spaces. Let $X$ be a metric space, $K \In X$ a subset. * sequential compactness: we say $K$ is compact, if every sequence in $K$ has a convergent subseq. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_15&rev=1771684872&do=diff Lecture 15 $\gdef\ucal{\mathcal U}$ Last time we discussed two versions of compactness: the sequential compactness and the open cover compactness. Theorem: let $X$ be a metric space. Then, $X$ is sequentially compact, if and only if $X$ is open cover compact. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_16&rev=1771684872&do=diff Lecture 16 connectedness Last time we didn't introduce the notion of a connected space. Let's do it now. Sometimes, we can tell from a glance, whether a space is connected or not: the interval $[0,1]$ is connected, the set $\{1,2,3\}$ (with induced topology from $\R$) is not (there are gaps between points). But, we also know that topological space can be weird. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_17&rev=1771684872&do=diff Lecture 17 Continuous Maps and Compactness, Connectedness Prop: If $f: X \to Y$ is continuous, and $K \In X$ is compact, then $f(K)$ is compact. Proof: any open cover of $f(K)$ can be pulled back to be an open cover of $K$, then we can pick a finite subcover in the domain, and the corresponding cover in the target forms a cover of $f(K)$. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_18&rev=1771684872&do=diff Lecture 18: Sequence of functions “How to measure the distance between two functions?” Sequence of functions Just as you can have a sequence * of number in $\R$ * of vectors in $\R^n$ * of points in a general metric space $X$. You can have a sequence of functions. $f_n(x)$ text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_19&rev=1771684872&do=diff note . text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:lecture_20&rev=1771684872&do=diff note text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:notes:start&rev=1771684872&do=diff Lecture Notes