Give an example of a continuous function $f: A \to \mathbb{R}$ on a subset A of $\mathbb{R}$ and a Cauchy sequence $x_n$ in A in which $f(x_n)$ is not a cauchy sequence in $\mathbb{R}$
2. Take $f0,1] \longrightarrow \mathbb{R}" alt="f0,1] \longrightarrow \mathbb{R}" /> given by $x \mapsto x^{-1}$ and take $x_n=n^{-1}$. Then $f(x_n)=n$ is certainly not Cauchy.
3. Crucial point- this sequence converges to a point that is NOT in A. If the $x_n$ converged to a point, a, in A, since f is continuous on A, $f(x_n)$ would have to converge to f(a), and so be Cauchy.