Math 409-300 — Summer 2014

Assignment 1 - Due Monday, June 9

• Read sections 1.1-1.6.
• Do the following problems.
  1. Section 1.2: 0, 3, 4(c), 6, 7(c), 10
     • Problem 1.2.0: Let $a,b,c,d$ be real and consider each of the statements below. Decide which are true and which are false. Prove the true ones and give counterexamoles for the false ones.
       1. If $a < b$ and $ c < d$ then $ a c > c d $.
       2. If $a \le b$ and $c > 1$, then $|a+c| \le |b+c|$.
       3. If $a \le b$ and $b \le a+c$, then $|a-b| \le c$.
       4. If $a < b-\varepsilon $ for all $\varepsilon > 0$, then $a < 0$.
     • The positive part of a real number $a$ is defined by \[ a^+ = \frac{|a|+a}{2} \] and the negative part by \[ a^- = \frac{|a|-a}{2} \]
       1. Prove that $a=a^+ - a^-$ and that $|a|=a^+ + a^-$.
       2. Prove that \[ a^+ = \left\{\begin{array}[ll] \\ a & a\ge 0\\ 0 & a < 0 \end{array}\right. \quad \text{and}\quad a^- = \left\{\begin{array}[ll] \\ 0 & a\ge 0\\ -a & a < 0 \end{array}\right. \]
     • Problem 1.2.4(c): Solve for all $x \in \mathb R$: $|x^3 - 3x + 1| < x^3$
     • Problem 1.2.6: The arithmatic mean of $a, b\in \mathbf R$ is $A(a,b) = \frac{a+b}{2}$ and the geometric mean of $a,b \in [0,\infty)$ is $G(a,b) = \sqrt{ab}$. If $0 \le a \le b$, prove that $a \le G(a,b) \le A(a,b)$, and also prove that $G(a,b) = A(a,b)$ if and only if $a=b$.
     • Problem 1.2.7(c): Prove that $-3\le x \le 2$ implies that $|x^2+x-6| \le 6|x-2$.
     • Problem 1.2.10: For all $a,b,c,d\in \mathbf R$, prove that $(ab+cd)^2 \le (a^2+c^2)(b^2+d^2)$.
  2. Show that ${\mathbb Z}_2$ (integers mod 2) is a field.
  3. Show that ${\mathbb Z}_4$ is not a field.

Assignment 2 - Due Monday, June 16

• Read sections 2.1-2.4.
• Do the following problems.
  1. Section 1.3: 0(a,b), 1(a,b,e) (Just state the answers.), 2, 6(a), 7(a), 11
  2. Section 1.5: 0(d), 2(a,d)
  3. Use your favorite software to plot the following sequences:
     1. $x_n = 1/2^n$, $n=1$ to $6$.
     2. $x_n= \frac{1-(-1)^n}{n+1}$, $n=1$ to $9$.
     3. $x_n=1+ \frac{(-1)^n}{n}$, $n=1$ to $10$.
     4. $x_n=(-1)^n+\frac{1}{n}$, $n=1$ to $10$.

Assignment 3 - Quiz, Friday, 6/20/2014

• Read sections 2.1-2.4.
• Know the definitions and be able to state and/or prove the theorems.
  1. Definitions:
     1. Supremum and infimum of a set $E\subset \mathbb R$.
     2. Completeness axiom.
     3. Sequence, subsequence, limit of a sequence, bounded sequence.
  2. Theorems:
     1. Squeeze theorem (2.9(a)).
     2. Be able to prove Theorem 2.11. (Statement will be given.)
     3. Comparison theorem (2.17).

Assignment 4 - Due Friday, June 27

• Read sections 3.1-3.4.
• Do the following problems.
  1. Section 2.3: 2, 7
  2. Section 2.4: 2, 6
  3. Section 3.1: 6, 7, 8

Assignment 5 - Due Wednesday, July 9

• Read sections 4.1-4.3.
• Do the following problems.
  1. Section 3.2: 0(d), 2(d), 5
  2. Section 3.3: 0(a), 1(d), 2(a), 4, 5
  3. Section 3.4: 0(b), 1(b), 4(b), 8

Updated 6/27/14 (fjn).