1. ## not cauchy

Give an example of a continuous function $\displaystyle f: A \to \mathbb{R}$ on a subset A of $\displaystyle \mathbb{R}$ and a Cauchy sequence $\displaystyle x_n$ in A in which $\displaystyle f(x_n)$ is not a cauchy sequence in $\displaystyle \mathbb{R}$

2. Take $\displaystyle f0,1] \longrightarrow \mathbb{R}$ given by $\displaystyle x \mapsto x^{-1}$ and take $\displaystyle x_n=n^{-1}$. Then $\displaystyle f(x_n)=n$ is certainly not Cauchy.

3. Crucial point- this sequence converges to a point that is NOT in A. If the $\displaystyle x_n$ converged to a point, a, in A, since f is continuous on A, $\displaystyle f(x_n)$ would have to converge to f(a), and so be Cauchy.