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Math 3001-001: Analysis 1, Fall 2019


Homework



Assignment
Assigned
Due
Problems
HW1 9/4/18
9/11/18
Read Sections 1.1-1.3.

1. Do Exercise 1.2.4.

2. Do Exercise 1.2.7(b) and 1.2.7(c).

3. Do Exercise 1.2.8.

Solution sketches.

HW2 9/11/18
9/18/18
Read Sections 1.4-1.5.

Read the following problems, but do not turn them in: 1.3.3, 1.3.6(a)(b)(c), 1.4.3, 1.4.8.

1. Do Exercise 1.3.2.

2. Do Exercise 1.3.9.

3. Does the non-Archimedean field $\mathbb R(t)$ satisfy the Nested Interval Property? Explain.

Solution sketches.

HW3 9/18/18
9/25/18
Read Sections 1.5-1.6.

Read the following problems, but do not turn them in: 1.5.4, 1.5.5, 1.5.9, 1.6.10.

1. Do Exercise 1.5.2.

2. Do Exercise 1.5.8.

3. Show that if $S\subseteq [0,1]$ is uncountable, then there is a real number $r\in[0,1]$ such that both $[0,r]\cap S$ and $[r,1]\cap S$ are uncountable.

Solution sketches.

HW4 9/25/18
New Due Date!
10/4/18

Read Sections 2.1-2.4.

1. Do Exercise 2.2.4.

2. Do Exercise 2.3.7 (a), (b), (c).

3. Do Exercise 2.4.4 (a).

Solution sketches.

HW5 10/2/18
10/9/18 Read Sections 2.5-2.6.

1. Do Exercise 2.4.2. (Hint for (b): use MCT to prove $(y_n)$ converges.)

2. Do Exercise 2.4.3 (a). (Hint: use MCT to prove the sequence converges. To find the limit, show that it must satisfy $L^2-2=L$ and $L>0$.)

3. Do Exercise 2.5.1 (a), (b), (c).

Solution sketches.

HW6 10/16/18
10/23/18 Read Section 2.7.

1. Do Exercise 2.6.2 (a) (c).

2. Do Exercise 2.6.4 (c). (Where the author says ``Decide whether each of the following sequences is a Cauchy sequence'', interpret this to mean ``Decide whether each of the following sequences MUST BE a Cauchy sequence''.)

3. Do Exercise 2.7.4 (a) (d).

Solution sketches.