Lecture Notes math104-f21

Equivalences of two definitions of compactness

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

Equivalences of two definitions of compactness $\gdef\In\subset$ Let $(X,d)$ be a metric space, $K \In X$ a subspace. Recall that we had the following definitions: (Compactness) : $K$ is compact if for any open cover of $K$, there exists a finite sub-cover.

math104-f21:final-grades

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

Each individual grade is posted on b-course. If you would like to see your exam sheet, please email me, I will send you a picture of your exam file. If there are no questions about the final grade, the overall grade will be submitted on Friday evening.

Common Mistakes in Final

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

Common Mistakes in Final 1 In general, we don't have $a^n + b^n \neq (a+b)^n $. 2 Given $\sum_n a_n$ converge, and $a_n > 0$ * It is wrong to conclude that $\limsup (a_{n+1}/a_n)) < 1$. * It is wrong to conclude $a_n$ is decreasing. 3 One need to show that $x_n$ is bounded from below; $x_n$ is monotone decreasing. These two conditions together show $x_n$ converges to some $x$. Then one need to prove that $x=\sqrt{a}$. Missing any of the three steps would make one lose points.

Heine-Borel theorem

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

Heine-Borel theorem $\gdef\In{\subset}$ The theorem states that, for the metric space $\R^n$, a subset is compact if and only if it is closed and bounded. We will prove this theorem for $\R$ first, then we show that if $A \In X$ is compact and $B \In Y$ is compact, then $A \times B \In X \times Y$ is compact.

HW 1 (with solution)

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 1 (with solution) Due Tuesday (Aug 31) 6pm. 2 points each. 1. Someone claims that he has found a smallest positive rational number, but would not tell you which number it is, can you prove that this is impossible? (Optional extra question: can you prove that there is no smallest rational number among all rational numbers that are larger than $\sqrt{2}$?)

HW 2 (with Solution)

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 2 (with Solution) Due Thursday Sep 9th, 6pm. In the following, a sequence $(a_n)$ means $(a_n)_{n=0}^\infty$, unless otherwise specified. You can only use properties of real number proved in Tao's book, section 5.4. 1. Let $(a_n)$ be a sequence in $\Q$. Suppose there is a rational $0 < r < 1$, such that $|a_{n+1} - a_n| < r |a_n - a_{n-1}|$, prove that $(a_n)$ is a Cauchy sequence.

HW 2

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 2 Due Thursday Sep 9th, 6pm. In the following, a sequence $(a_n)$ means $(a_n)_{n=0}^\infty$, unless otherwise specified. You can only use properties of real number proved in Tao's book, section 5.4. 1. Let $(a_n)$ be a sequence in $\Q$. Suppose there is a rational $0 < r < 1$, such that $|a_{n+1} - a_n| < r |a_n - a_{n-1}|$, prove that $(a_n)$ is a Cauchy sequence.

HW 3 with solution

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 3 with solution * [Tao] Ex 5.4.3, * [Tao] Ex 5.4.5 (you may assume result in 5.4.4) * [Tao] Ex 5.4.7, * [Tao] Ex 5.4.8 * Let $A$ be the subset of $\Q$ consisting of rational numbers with denominators of the form $2^m$. Prove that for any $x \in \R$, there is a Cauchy sequence $(a_n)$ in $A$, such that $x = LIM a_n$.

HW 4

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 4 1. If $x \in \R$ and there is a Cauchy sequence $(a_n)$ in $\Q$ such that $x = LIM a_n$, then show that $x = \lim a_n$. 2. For any $a \in \R$, prove that $\lim a^n / n! = 0$. 3. Let $A = \{ q \in \Q: q^2 < 3 \}$, let $x = \sup(A)$, prove that $x^2 = 3$.

HW 5

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 5 0. Correct your mistakes in midterm 1, you don't need to submit the correction. 1. Prove that there is a sequence in $\R$, whose subsequential limit set is the entire set $\R$. 2. Prove that $\limsup(a_n+b_n) \leq \limsup(a_n) + \limsup(b_n) $, where $(a_n)$ and $(b_n)$ are bounded sequences in $\R$.

HW 6

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 6 Due next Thursday, 10/7, 6pm 1. Ross Ex 14.1 (briefly describe your reasoning) 2. Ross Ex 14.4 3. Let $\sum_{n=1}^\infty a_n$ be a series. Show that if $\sum_{m=1}^\infty a_{2m}$ and $\sum_{m=1}^\infty a_{2m-1}$ both converges, then $\sum_{n=1}^\infty a_n$ converges.

HW 7

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 7 Metric space and topology. $\gdef\In{\subset}$ 1. Consider the metric space $(X = \Z, d_X(x,y) = |x-y|)$. Write down the open ball $B_3 (2)$, $B_{1/2}(2)$. Is the subset $\{2\}$ open in $X$? Is it closed in $X$? Explain. 2. Consider the metric space $(X = [0,1) \In \R, d_X(x,y) = |x-y|)$. Write down the open ball $B_{3} (0)$, $B_{1/3}(0)$. Is the subset $[0,1)$ open in $X$? closed in $X$? Is the subset $[1/3, 1)$ open in $X$? closed in $X$?

HW 8

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 8 $\gdef\In{\subset}$ 1. Determine whether following subset $S$ of metric space $X$ is (a) open or not (b) closed or not (c ) bounded or not (d) compact or not. (You may use Heine-Borel theorem for $\R^k$) * $X = \R$ with usual metric. $S = \Q \cap [0,1]$.

HW 9

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 9 In the following, all the subsets of $\R$, or $\R^2$, are endowed with the induced topology. 1. Let $\Z \subset \R$ and $Y$ any topological space. Prove that any map $f: \Z \to Y$ is continuous. 2. Let $X = [0, 2\pi ) \subset \R$, and $Y \subset \R^2$ be the unit circle. Let $f: X \to Y$ be given by $f(t) = (\cos(t), \sin(t))$. Prove that $f$ is a bijection and continuous, but $f^{-1}: Y \to X$ is not continuous. (Remark: To show $f$ is a bijection and continuous, you may consider $F:\…

HW 10

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 10 We will use the following notion of equivalence of metrics. Let $X$ be a set, and let $d_1$, $d_2$ be two distance functions on $X$, such that $(X, d_1)$ and $(X,d_2)$ are both metric spaces. We say $d_1$ and $d_2$ are equivalent if there exists positive constants $c_1, c_2$, such that for any $p,q \in X$, we have $$ c_1 d_1(p,q) \leq d_2(p,q) \leq c_2 d_1(p,q). $$

HW 11

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 11 For the following question: if true, prove it; if false, give a counter-example. 1. Assume $f: X \to Y$ is uniformly continuous, and $g: Y \to Z$ is uniformly continuous. Is it true that $g \circ f: X \to Z$ is uniformly continuous? 2. If $f: (0,1) \to (0,1)$ is a continuous map, is it true that there exists a $x \in (0,1)$, such that $f(x) = x$? What if one change the setting to $f: [0,1] \to [0,1]$ being continuous, does your conclusion change?

HW 13

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 13 Due Monday (Nov 29) 9pm. 8-O problem 6 contains a typo, and it is updated now. 1. In class we have seen that a function $f(x)$ may be differentiable everywhere, but the derivative function $f'(x)$ is not continuous. In this problem, we will see that discontinuity is not a removable singularity. Assume $f: \R \to \R$ is differentiable. Assume that $\lim_{x \to 0} f'(x) = 1$. Prove that $f'(0)=1$. (Hint, you can use mean value theorem, and definition of the $f'(0)$. )

HW 15

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

HW 15 This is for your practice only, not going to be graded. The solution will be release next Wednesday. 1. Ross Ex 33.9 (uniform convergence and integral). See Rudin Thm 7.16 (page 151) 2. Ross Ex 33.5 (bound an integral by replacing an part of the integrand by something nice)

Midterm 1: Review

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

Midterm 1: Review In the first part of this course, we covered the construction of real number, and some results about limit. Here is a list of key concepts * The numbers $\N, \Z, \Q$. * The axioms of field, an example of finite field $\F_5$.

Midterm 2 Review

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

Midterm 2 Review This midterm will have 20% on series and 80% on metric space topology and continuous functions. For series, you need to know the basic definitions of convergence, absolute convergence. And various examples that illustrates the differences between these notions. Root test and ratio test for absolute convergence. Read Ross for a good account of examples.

math104-f21:midterm_1

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

[ midterm 1 and solution] Exam stats: 36 people taking exam. Average score is 29 points out of total 50 points. * 40 - 50: 11 ppl * 30 -29: 8 * 20-29: 8 * below 15: 9

Midterm 2

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

Midterm 2 [midterm 2 exam and solution] Stat range # of people 40-50 12 30-39 5 20-29 6 10-19 7 0-9 2 Average: 30.

Math 104: Introduction to Real Analysis (2021 Fall)

Anonymous (anonymous@undisclosed.example.com) — 2026-02-21T14:41:08+00:00

Math 104: Introduction to Real Analysis (2021 Fall) $$\gdef\Q{\mathbb{Q}}$$ Instructor: Peng Zhou Email: pzhou.math@berkeley.edu Office: Evans 931 Office Hour: Monday 12:10-1pm, updated Wednesday 10:10-11am, Friday 10:10-11am Lecture: MWF, 11:10am - 12:00. Etcheverry 3107.