A question from a test I had last week that I failed to answer has been bothering me:
Let $A$ be a non-empty part of $\mathbb{R}$ and $f: A \rightarrow \mathbb{R}$ a function. Investigate the relation between the following statements:
a. For all $M \in \mathbb{R}^+$ is the limitation of $f$ to $A \cap [-M, M]$ uniformly continuous.
b. When $(x_n)_{n \in \mathbb{N}}$ is a Cauchy sequence in $A$, then $f(x_n))_{n \in \mathbb{N}}$ is a Cauchy sequence in $\mathbb{R}$.
I know how to prove a $\rightarrow$ b, but I can't quite prove nor disprove the converse. I found this link that discusses a similar problem, but it only seems to provide an answer in the case of compact completion, i.e. when the interval is closed. However, if you take for example, $A = (0,1)$ then $A \cap [-M, M]$ is always half open (or containing just $0$), and therefore not always complete. What do I do to find a proof in this case?
Any help is greatly appreciated!