https://courses.pzhou.org/ 2026-04-17T11:20:29+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:hw:hw1&rev=1771684872&do=diff HW1 Due on gradescope on Jan 28 6pm. 0. Join our discord server, and setup your own homepage in the Math 104, Students Area. 1. Try these exercises Ross 1.10, 1.12, 2.1, 2.2, 2.7, 3.6, 4.11, 4.14, 7.5 Try not to look up the solutions. And write up the argument in your own way, and try to be clear make your writing easy to follow. It is OK that you don't completely solve the problems, but do try to give each some thoughts. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw2&rev=1771684872&do=diff HW2 Due on gradescope next Friday 6pm. You should also submit on discord around Wednesday for others to comment on it. 0. Discussion problems: 9.9, 9.15, 10.7, 10.8 1. About recursive sequence: Ross Ex 10.9, 10.10, 10.11 2. Squeeze test. Let $a_n, b_n, c_n$ be three sequences, such that $a_n \leq b_n \leq c_n$, and $L = \lim a_n = \lim c_n$. Show that $\lim b_n = L$. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw3&rev=1771684872&do=diff HW 3 * Ross Ex 10.6, 11.2, 11.3, 11.5 * How would you explain 'what is limsup'? For example, you can say something about: What's the difference between limsup and sup? What is most counter-intuitive about limsup? Can you state some sentences that seems to be correct, but is actually wrong? text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw4&rev=1771684872&do=diff HW4 0. List $\geq 5$ concrete/detailed questions that you wanted to ask (but were too afraid to ask), about lecture notes, about homework, about textbook statements or proofs. Post them on your course homepage, then post the link to discord. Try help others text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw5&rev=1771684872&do=diff HW 5 1-3. Ross 13.3, 13.5, 13.7 4. Recall that in class, given $(X, d)$ a metric space, and $S$ a subset of $X$, we defined the closure of $S$ to be $$ \bar S = \{ p \in X \mid \text{there is a subsequence $(p_n)$in $S$ that converge to $p$\} $$ text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw6&rev=1771684872&do=diff HW6 This week we learned about continuous functions, and compactness. We will have some homework about proving compactness. 1. In class, we proved that $[0,1]$ is sequentially compact, can you prove that $[0,1]^2$ in $\R^2$ is sequentially compact? (In general, if metric space $X$ and $Y$ are sequentially compact, we can show that $X \times Y$ is sequentially compact.) text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw7&rev=1771684872&do=diff HW 7 This week we proved the equivalence of the two notions of compactness. Here are some more problems 1. If $X$ and $Y$ are open cover compact, can you prove that $X \times Y$ is open cover compact? (try to do it directly, without using the equivalence between open cover compact and sequential compact) text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw8&rev=1771684872&do=diff HW 8 Let's consider a few examples of sequences and series of functions. 1. Let $f_n(x) = \frac{n + \sin x} {2n + \cos n^2 x}$, show that $f_n$ converges uniformly on $\R$. 2. Let $f(x) = \sum_{n=1}^\infty a_n x^n $. Show that the series is continuous on $[-1, 1]$ if $\sum_n |a_n| < \infty$. Prove that $\sum_{n=1}^\infty n^{-2} x^n$ is continuous on $[-1, 1]$. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw9&rev=1771684872&do=diff HW 9 The last two weeks, we studied derivation, with topics like * mean value theorem * intermediate value theorem * Taylor expansion Exercises: * Read Ross p257, Example 3 about smooth interpolation between $0$ for $x \leq 0$ and $e^{-1/x}$ for $x>0$. Construct a smooth function $f: \R \to \R$ such that $f(x)=0$ for $x\leq 0$ and $f(x)=1$ for $x\geq 1$, and $f(x) \in [0,1]$ when $x \in (0,1)$. text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw10&rev=1771684872&do=diff HW 10 Ross 33.4, 33.7, 33.13, 35.4, 35.9(a) text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:hw11&rev=1771684872&do=diff HW 11 Ross 34.2, 34.5, 34.7 ---------- Optional: Rudin: Ex 15 (Hint: use 10( c ) ), 16 and an extra one: Let $f:[0,1] \to \R$ be given by $$ f(x) = \begin{cases} 0 &\text{if } x = 0 \cr \sin(1/x) &\text{if } x \in (0,1] \end{cases}. $$ And let $\alpha: [0, 1] \to \R$ be given by $$ \alpha(x) = \begin{cases} 0 &\text{if } x = 0 \cr \sum_{n \in \N, 1/n<x} 2^{-n} &\text{if } x \in (0,1] \end{cases}. $$ Prove that $f$ is integrable with respect to $\alpha$ on $[0,1]$. Hint: prove that $… text/html 2026-02-21T14:41:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math104-s22:hw:start&rev=1771684872&do=diff Homeworks