1. See the book for definitions/statements.

    2.  
  2. For each description, either give an example that fits the description or explain why it is impossible to do so.

    3.  
    1. A sequence MATH and a subsequence MATH with $\lim s_{n}=2$ and $\lim s_{n_{k}}=-2.$

      2. MATH
       
    2. A sequence MATH and two subsequences MATH and MATH with $\lim s_{n_{k}}=1$ and $\lim s_{m_{k}}=3.$

      3. MATH
       
    3. A pair of sequences MATH and MATH so that $s_{n}<t_{n}$MATH and MATH

      4. MATH
       
  3. Prove (directly from the definition of diverging to $\infty $) that MATH

    4. MATH
     
  4. Let $s_{1}=1.$$\ $Let $0<a<1.$ Let $s_{n+1}=as_{n}.$
    1. Show that MATH

      2. MATH
       
    2. Show that MATH is monotone.

      3. MATH
       
    3. Explain why $\lim s_{n}$ exists and find $\lim s_{n}.$

      4. MATH
       
  5. Prove the following: If MATH is a convegent sequence then it is a Cauchy sequence. (Hint: Write MATH as MATH where $L$ is an appropriately chosen value.)

    6. MATH
     
  6. For each of the following sequences $\left( i\right) $ Determine if it is monotone $\left( ii\right) $ Determine if it is bounded $\left( iii\right) $ Find $\lim \inf s_{n}$ and $\lim \sup s_{n}$$\left( iv\right) $ Find $\lim s_{n},$ if it exists.
    1. MATH

      2. MATH
       
    2. MATH

      3. MATH

The following 2 problems are the Take-Home portion of Exam 2.

  1. Let MATH and MATH be sequences such that $\lim s_{n}=\infty $ and MATH Prove that MATH

    2. MATH
     
  2. Prove (directly from the definition of a Cauchy sequence) that MATH is Cauchy (i.e., do not use any theorems about Cauchy sequences.)

    3. MATH
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