Exercise 2.6.2

Question

Give an example of each of the following, or argue that such a request is impossible.
  \begin{enumerate}[label=(\alph*)] 
  \item A Cauchy sequence that is not monotone.
  \item A Cauchy sequence with an unbounded subsequence.
  \item A divergent monotone sequence with a Cauchy subsequence.
  \item An unbounded sequence containing a subsequence that is Cauchy.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item $x_n = (-1)^n/n$ is Cauchy by Theorem 2.6.2.
  \item Impossible since all Cauchy sequences converge and are therefore bounded.
  \item Impossible, if a subsequence was Cauchy it would converge, implying the subsequence would be bounded and therefore the parent sequence would be bounded (because it is monotone) and thus would converge.
  \item $(2, 1/2, 3, 1/3, \dots)$ has subsequence $(1/2, 1/3, \dots)$ which is Cauchy.
   \end{enumerate}

Exercise 2.6.2

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Exercise 2.6.2

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