Recall that $(y_{n})_{n}$ is a convex block subsequence of a sequence $(x_{n})_{n}$ in a Banach space $X$ provided that there exists a strictly increasing sequence of positive integers $(k_{n})_{n}$ so that $y_{n}\in \textrm{co}(x_{i})_{i=k_{n-1}+1}^{k_{n}}$ for every $n$ ($k_{0}=0$). The collection of all convex block subsequences of $(x_{n})_{n}$ will be denoted by $\textrm{cbs}((x_{n})_{n})$.

Let $(x_{n})_{n}$ be a bounded sequence in a Banach space $X$. We set $$\textrm{ca}((x_{n})_{n})=\inf_{n}\sup_{k,l\geq n}\|x_{k}-x_{l}\|.$$ Then $(x_{n})_{n}$ is norm-Cauchy if and only if $\textrm{ca}((x_{n})_{n})=0$.

Let $X$ be a Banach space. We set $$R(X)=\sup_{(x_{n})_{n}\subseteq B_{X}}\inf_{(z_{n})_{n}\in \textrm{cbs}((x_{n})_{n})}\textrm{ca}((z_{n})_{n}).$$

We have proved that a Banach space $X$ is reflexive if and only if $R(X)=0$.

Furthermore, we have shown that $R(X)=2$ if $X$ contains a subspace isomorphic to $l_{1}$ (not contains $l_{1}$ as a subset), and $R(c)=2$, where $c$ denotes the space of all convergent scalar sequences equipped with the supremum norm. My concern is $R(c_{0})$.

Question. Is $R(c_{0})\leq 1$ ?

Thank you.

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