Let $(X,d)$ be a metric space and let $(x_n)_{n\in\mathbb{N}}$ be a Cauchy sequence in $X$, i.e. $d(x_n,x_m)$ goes to $0$ when $n,m\rightarrow\infty$. The sequence does not necessarily have a limit in $X$, however.
I'm wondering if for fixed $k$, the sequence $d(x_k,x_l)$ has a limit in $\mathbb{R}$ when $l\rightarrow\infty$? I know that $\lim\sup_{l\to\infty}d(x_k,x_l)$ exists (this is always true for Cauchy sequences), but what about the limit?
Thank you very much in advance 🙂
Best Answer
Given any point $w\in X$ the function "distance from $w$" $$ d(w,\cdot):X\longrightarrow\Bbb R $$ is continuous and transforms Cauchy sequences (in $X$) into Cauchy sequences (in $\Bbb R$). But $\Bbb R$ is complete!