Exercise 4.4.6

Question

Give an example of each of the following, or state that such a request is impossible. For any that are impossible, supply a short explanation for why this is the case.
  \begin{enumerate}[label=(\alph*)] 
  \item A continuous function $f:(0,1) \rightarrow \mathbf{R}$ and a Cauchy sequence $\left(x_{n}\right)$ such that $f\left(x_{n}\right)$ is not a Cauchy sequence;
  \item A uniformly continuous function $f:(0,1) \rightarrow \mathbf{R}$ and a Cauchy sequence $\left(x_{n}\right)$ such that $f\left(x_{n}\right)$ is not a Cauchy sequence;
  \item A continuous function $f:[0, \infty) \rightarrow \mathbf{R}$ and a Cauchy sequence $\left(x_{n}\right)$ such that $f\left(x_{n}\right)$ is not a Cauchy sequence;
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item $f(x) = 1/x$ and $x_n = 1/n$ has $f(x_n)$ diverging, hence $f(x_n)$ is not Cauchy.
  \item Impossible since for all $\epsilon > 0$ we can find an $N$ so that all $n \ge N$ has $|x_n - x_m| < \delta$ (since $x_n$ is Cauchy) implying $|f(x_n) - f(x_m)| < \epsilon$ and thus $f(x_n)$ is Cauchy. (Uniform continuity is needed for the $\forall n \ge N$ part)
  \item Impossible since $[0,\infty)$ is closed $(x_n) 	o x \in [0,\infty)$ implying $f(x_n) 	o f(x)$ since $f$ is continuous, thus $f(x_n)$ is a Cauchy sequence.
   \end{enumerate}

Exercise 4.4.6

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Exercise 4.4.6

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