show that $\lim_{n→\infty} f(x_n)$ exists.

Question

Let $A \subset \Bbb{R}^n$ and $f : A \longrightarrow \Bbb{R}^m$ be a uniformly continuous function. If ${x_n}$ where $n\geq 1$ is a Cauchy sequence in $A$ then show that $\lim_{n\to\infty}f(x_n)$ exists. If the domain be $\mathbb{R}$ then there is no problem to solve it but I need help here. Please help me to solve it.

Answer 1

Hint$\;\;$ Show that the sequence $\,\{f(x_n)\}\;$ is Cauchy and hence converges, since $\,\mathbb{R}^m\,$ is complete.