Syllabus
Book we follow (free!): "Primer of Real Analysis" by Dan Sloughter
Student produced flashcards for definitions
Homework
For Exam 1 (final revisions due 14 September at 11:59PM)
HW1 (due Monday 26 August) [soln]: Exercises 1.1.3, 1.1.4, 1.3.3, 1.3.4 (b-d), 1.3.6, 1.3.8, 1.3.11
HW2 (due Monday 2 September) [soln]: Exercise 1.3.3, 1.3.4 (b-d), 1.3.6, 1.3.8, 1.3.11, 1.4.6(b,c), 1.4.9, 1.4.10, 2.1.1, 2.1.2, 2.1.6, 2.1.7, 2.1.10, 2.1.12, 2.1.14, 2.1.17, 2.1.18, 2.1.19, 2.1.21, 2.1.23, 2.1.24, 2.1.26
HW3 (due Friday 6 September Saturday 7 September) [soln]: 1.3.10, 1.3.11, 1.4.1, AND write a proof that $\displaystyle\lim_{n \rightarrow \infty} \dfrac{n+3}{n+2} = 1$. (on last one, you can "work backwards" as I showed in the lecture to find the $N$ that you need)
For Exam 2
HW4 (due WednesdayThursday 18 September) [soln]: Exercises 2.1.1, 2.1.2, 2.1.3
HW5 (due Monday 23 September) [soln]: Exercises 2.1.6(c), 2.1.7, 2.1.8, 2.1.10, 2.1.11
HW6 (due Monday 30 September) [soln]: Exercises 2.1.12, 2.1.13, 2.1.14, 2.1.15, 2.1.16, 2.1.17, 2.1.18, 2.1.19, 2.1.21, 2.1.22, 2.1.23, 2.1.24
HW7 (due Monday 7 October) [soln]: Exercises 2.1.25, 2.1.26, 2.1.27, 2.2.1, 2.2.2, 2.2.4, 2.2.5, 2.2.6
HW8 (due >FridaySaturday 1112 October) [soln]: Exercises 2.2.7, 2.2.9, 2.2.10, 4.2.1, 4.2.2, 4.2.3
For Exam 3
HW9 (due Wednesday 23 October) [soln]: 4.2.1, 4.2.2, 4.2.3
HW10 (due Monday Tuesday 2829 October) [soln]: Exercises 4.3.2, 4.3.5, 4.3.6, 4.3.7, 4.3.8
HW11 (due Monday 4 November):
HW12HW11 (due Friday 8 November) [soln]: Exercises 4.3.9, 4.3.10, 4.3.11, 4.4.2, 4.4.3, 4.4.4, 4.4.5, 4.4.6, 4.4.7
For Final Exam
HW12 (due Wednesday 27 20 November) [soln]: Exercises 5.1.1, 5.1.2, 5.1.3, 5.1.4, 5.1.5, 5.1.6, 5.1.7, 5.1.8
HW13 (due Monday 2 December) [soln]: Exercises 5.1.11, 5.3.1, 5.3.2, 5.3.3, 5.3.4, 5.4.1, 5.4.2, 5.4.3, 5.4.5, 5.4.7, 5.4.8, 5.4.9, 5.4.10, 5.4.11, 5.4.12, 5.4.13, 5.4.14
HW14 (due Friday 6 December) [soln]: Exercises 5.4.18, 5.4.19, 5.4.23, 5.4.24, 5.4.25, 5.4.26
Quizzes
Quiz 1 (due 23 Aug) [soln]: Let $A=\left\{ \text{green}, *\right\}$ and $B=\left\{A,4\right\}$. Compute $A \cup B$, $A \cap B$, $A \times B$, and $B \times A$.
Quiz 2 (due 23 Aug) [soln]: Show that the relation $R$ on $\mathbb{Z}$ defined by $m \sim_R n$ iff $m-n$ is even is a transitive relation.
Quiz 3 (due 27 Aug) [soln]: Show that for all $a,b,c \in \mathbb{Q}$ that $a(b+c)=ab+ac$ (and cite relevant equation numbers).
Quiz 4 (due 29 Aug) [soln]: Turn the proof sketch in the notes (see teams) that it is impossible for both $a>0$ and $a=0$ in $\mathbb{Q}$ into a nice proof.
Quiz 5 (due 2 Sep 3 Sep) [soln]: Turn the proof sketch in the notes (see teams) that $|a-b| \geq 0$ into a nice proof.
Quiz 6 (due Sat 7 Sep) [soln]: Consider the sequence $a_n=\dfrac{2n+1}{n+2}$. Prove that $\displaystyle\lim_{n \rightarrow \infty} a_n=2$. In your proof, you can choose $N$ so that $N+2 > \dfrac{3}{\epsilon}$.
Quiz 7 (due 17 Sep) [soln]: Find $\displaystyle\limsup_{n\rightarrow \infty} \dfrac{1}{n}$ and $\displaystyle\liminf_{n\rightarrow\infty} \dfrac{1}{n}$.
Quiz 8 (due 25 Sep) [soln]: Show that the sequence $a_n=\dfrac{(-1)^n}{n}$ is a Cauchy sequence.
Quiz 9 (due 1 Oct) [soln]: Write a clean proof of the theorem that if $k \geq m$ and $\displaystyle\sum_{i=m}^{\infty} a_i$ converges, then $\displaystyle\sum_{i=k}^{\infty} a_i$ converges that was shown in the 30 September class notes.
Quiz 10 (due 14 Oct) [soln]: Prove that the constant sequence $a_n=c$ has limit $\displaystyle\lim_{n\rightarrow\infty} a_n=c$.
Quiz 11 (due 21 Oct) [soln]: Exercise 4.3.1 (a) and (c)
Quiz 12 (due 24 Oct) [soln]: Let $I_n=\left[ 1-\dfrac{n}{n+1}, 1+\dfrac{n}{n+1}\right]$ for $n=1,2,3,\ldots$. Are the following sets open, closed, or neither?:
(a) $\displaystyle\cap_{n=1}^{\infty} I_n$ and (b) $\displaystyle\cup_{n=1}^{\infty} I_n$?
Quiz 13 (due 31 Oct) [soln]: Find an open cover of the open set $(0,1) \cup (2,3)$ that does not have a finite subcover.
Exams
Exam 1
Exam 2
Exam 3
