1. The definitions and statments can be found in the book. (See the index for the page numbers).

    2.  
  2. Determine if each statement is true or false. If it is true, write TRUE and explain how you determined this. If it is false, write FALSE and provide a counterexample.
    1. If $a,b,c\in R$ then MATH

      2. MATH
       
    2. If $\sup S<\sup T,$ then $s\leq t$ for all $s\in S$ and $t\in T.$

      3. MATH
       
    3. If MATH then $\lim s_{n}=2.$

      4. MATH
       
  1. Let $S\subseteq R$ be a bounded, nonempty set and let $a\in \QTR{Bbb}{R}.$ If MATH prove that $\sup M=a+\sup S.$

    2. MATH

    MATH
     

  2. Prove (formally, i.e., using $\varepsilon $) that MATH

    3. MATH
     
  3. Let MATH Find each of the following quantities if it exists.
    1. $\max S.$

      2. MATH
       
    2. $\min S.$

      3. MATH
       
    3. $\sup S.$

      4. MATH
       
    4. $\inf S.$

      5. MATH
       
  4. Prove the following directly from the Field and Order properties (on the cover sheet) and the properties 
    X1: MATH for all $a,b$ and X2: $-0=0.$
    1. If $a\leq b,$ then $-b\leq -a.$

      2. MATH

      MATH
       

    2. If $a\leq b$ and $c\leq 0,$ then $bc\leq ac.$

      3. MATH
       
       
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