Math 409-300 — Summer 2015

Assignment 1 - Due Monday, June 8, 2015

• 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 counterexamples 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 $x \in \mathbb R$: $|x^3 - 3x + 1| < x^3$

     • 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 $a,b,c,d\in \mathbb R$, 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 Friday, June 12, 2015

• Read sections 2.1-2.4.
• Do the following problems.
  1. Section 1.3: 0(a,b), 1(a,d,e), 2, 5, 7, 10
  2. Let $E$ be a bounded subset of $\mathbb R$ and let $U$ be the set of all upper bounds for $E$. Show that $U$ is bounded below, that $s=\inf(U)=\sup(E)$, and that $s \in U$.

Assignment 3 - Due Wednesday, June 17, 2015

• Read sections 2.1-2.4.
• Do the following problems.
  1. Section 1.3: 9,11
  2. Section 1.4: 1(d), 4(c), 5, 8
  3. Section 1.5: 1(a)(β,ε), 1(b)(β,ε), 2(d)
  4. (Bonus) Prove these:
     1. (15 pts.) If $\ell \in \mathbb N$ then $\sum_{k=1}^n k^\ell = p_\ell(n)$, where $p_\ell$ is a polynomial of degree $\ell+1$.
     2. (10 pts.) If $\ell=3$, $p_3(n) = n^2(n+1)^2/4$.

Assignment 4 - Due Monday, June 22, 2015

• Read sections 2.3-2.4.
• Do the following problems.
  1. Section 2.1: 5, 6, 7(b)
  2. Section 2.2: 1(d), 2(c), 5, 8(b)
  3. Section 2.3: 2, 5

Assignment 5 - Due Friday, June 26, 2015

• Read sections 2.5, 3.1-3.4.
• Do the following problems.
  1. Section 2.4: 4, 5, 6
  2. Section 2.5: 1(b,c,e), 5(a)
  3. Section 3.1 1(b,d)

Assignment 6 - Due Wednesday, July 1, 2015

• 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

Updated 6/26/15 (fjn).