I tried the following exercise:
Give an example of each of the following or state that such a request is not possible. For any that are impossible supply an explanation for why this is the case.
(a) a continuous function $f: (0,1) \to \mathbb R$ and a Cauchy sequence $x_n$ such that $f(x_n) $ is not a Cauchy sequence
(b) a continuous function $f: [0,1] \to \mathbb R$ and a Cauchy sequence $x_n$ such that $f(x_n) $ is not a Cauchy sequence
(c) a continuous function $f:[0,\infty) \to \mathbb R$ and a Cauchy sequence $x_n$ such that $f(x_n)$ is not a Cauchy sequence
(d) a continuous bounded function $f$ on $(0,1)$ that attains a maximum value but not a minimum value (on $(0,1)$)
Please can you tell me if my anwer are correct?
(a) possible: $f(x) = 1/x$ and $x_n = 1/n$.
(b) not possible: $f$ is uniformly continuous. Let $\varepsilon > 0$. Let $\delta >0$ be such that $|x-y|<\delta$ imply $|f(x)-f(y)|<\varepsilon$. Then there is $N$ with $n,m>N$ imply $|x_n-x_m|<\delta$ imply $|f(x_n)-f(x_m)|<\varepsilon$
(c) like in (a): $f(x) = 1/x$ and $x_n = 1/n$.
(d) $f(x) = x$ if $x\in (0,1/2]$ and $f(x) =1-x$ if $x \in [1/2,1)$.
Best Answer
Given that you're there already, I'll try to add some context.
A continuous function maps convergent sequences to convergent sequences (as far as I know, this result on real functions is ascribed to Heine), and convergent sequences are always Cauchy. Therefore, $(f(x_n))_n$ might fail to be Cauchy only if it fails to converge, and that might occur only if $(x_n)$ fails to converge.
We therefore witness that the exercise is closely linked with the completeness of $f$'s domain. Perhaps the missing part is proving, generally, that whenever $A\subset\Bbb R$ isn't complete, there exists a continuous $f:A\to\Bbb R$ and a Cauchy sequence $(x_n)_n\subset A$ such that $(f(x_n))_n$ isn't Cauchy. That can be accomplished basically in the same manner you've managed part (a), by picking $x_0\in \overline{A}\setminus A$ and defining $f(x):=\frac{1}{x-x_0}$.
With regards to your questions in comments, I was indeed aiming at the fact that any Cauchy sequence in the domain converges (hence the sequence of images also converge, and is therefore Cauchy). In this, it might be easier to let the $\epsilon-\delta$ language rest, and rely on proven results which are more, for lack of a better word, illuminating in this case.
Without presuming to speak for him, I imagine @DavidMitra was hinting that you've basically reduced the problem to the one you dealt with in part (b), which is a different approach than what I had in minds (kudos, BTW).