Will a Cauchy sequence in $\mathbb{R}^n$ be a Cauchy sequence in $\mathbb{R}^{m}$?

Question

Assume we have a function $f: A \subseteq \mathbb{R}^n \to \mathbb{R}^m$ which is uniformly continuous. Will a Cauchy sequence in $A$ be mapped on a Cauchy sequence in $\mathbb{R}^{m}$?

I think this is true if $m < n$, but don't know how to prove it.

Thanks in advance

Answer 1

In general:

Let $X$ and $Y$ two normed spaces with norm $||\cdot||_A$ and $||\cdot||_B$ respectively. Let $f:A\rightarrow B$ is uniformly continuous. Now take $\{x_n\}$ a Cauchy sequence in $A$ and let $\epsilon>0$. So, exists $\delta>0$ such that $$ ||x-y||_A<\delta\Rightarrow||f(x)-f(y)||_B<\epsilon $$ Now, take $N\in\mathbb{N}$ such that $$ n,m\geq N\Rightarrow||x_n-x_m||_A<\delta $$ So $$ m,n\geq N\Rightarrow||f(x_n)-f(x_m)||_B<\epsilon $$ This means that $\{f(x_n)\}$ is Cauchy in $B$.