Understanding $l_p$ with the metric $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_k-y_k|}{1+|x_k-y_k|}$

Question

Consider a successions space $l_p$, for $p\geqslant1$ in which $x=(x_1,x_2,...,x_k...)$ and $\lim_{n\to\infty} x_n=x_0$. The metric defined is the following $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_k-y_k|}{1+|x_k-y_k|}$ Since the series converges $\forall \epsilon>0\exists\:n_0,\forall n\geqslant n_0$ $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_{n,k}-x_{0,k}|}{1+|x_{n,k}-x_{0,k}|}<\epsilon$.$k=1,2,3...$

Question:

I do not understand the $k$ in the following metric $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_{n,k}-x_{0,k}|}{1+|x_{n,k}-x_{0,k}|}$. Does the $k$ indicator implies the sequence (series) is defined on $\mathbb{R}^n$? Does the $k$ stands for each variable in which the sequence converges in the following way $|x_{n,k}-x_{0,k}|$ for each $k$? If not what is the meaning of $k$?

Answer 1

An answer to your questions is given by the accepted answer of the following question: The space of sequences as a complete metric space

You need to notice that there are two notion of convergence here: pointwise convergence, and the convergence of the sequence itself as an element of the $\ell_p$ space. That is: \begin{equation} x_n := \{x_{n,k}\}_{k=1}^\infty \in \ell_p \end{equation} $\{x_n\}$ is a sequence of sequences: $x_1$ is the first sequence, $x_2$ is the second ..etc each of them has infinite coordinates since they are elements of the space $\ell_p$.

Answer 2

Really I want us to look into it this way. Let

$ \textbf{x}=(x_{1},x_{2},...,x_{n}, ...)$ be a vector in an infinite dimension. The sequence of vector $ \textbf{x}$ can be denoted by $$ \{ \textbf{x}_k \} = \{ x_{n,k}\} = \{ x_{0,k},x_{1,k},...,x_{n,k},...\} $$ so, in the metric in question, $n$ is just a subscript denoting the $n^{th}$ component of the vector $\textbf{x}$. And $k$ is a subscript that denote the $k^{th}$ term in the sequence of vector $\textbf{x}_k$. In fact, $ |x_{n,k}-x_{0,k}|$ can be seen as sequence of distances between each term $x_{k,n}$ and a particular point $x_{0,k}$