Syllabus

Book we follow (free!): "Primer of Real Analysis" by Dan Sloughter

Student produced flashcards for definitions

HW1 (due Monday 26 August) [soln]: Exercises 1.1.3, 1.1.4,

HW2 (due Monday 2 September) [soln]: Exercise 1.3.3, 1.3.4 (b-d), 1.3.6, 1.3.8

HW3 (due

HW4 (due

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

HW9 (due

HW10 (due

HW12 (due

HW13 (due Monday 2 December):

HW14 (due

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

Quiz 6 (due

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.

Exam 1

Exam 2

Exam 3