# Questions tagged [weakly-cauchy-sequences]

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### About weakly Cauchy sequences in complete metric spaces

Let $(X,d)$ be a complete metric space. Call a sequence $(x_n)\subseteq X$ a weakly Cauchy sequence in $X$ if there is some $y\in X$ such that $(d(y,x_n))_n$ is a Cauchy sequence in $\mathbb{R}$. It ...
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### weakly cauchy sequence is norm compact

A sequence $(x_n)$ is weakly cauchy if for every $x^*\in X^*$, $(x^*(x_n))$ converges. Let $c$ denote the space of convergent functions. Theorem: A weakly cauchy sequence is norm-bounded. ...
Let $X$ be a Banach space and let us consider the linear subspace of $\ell_\infty(X)$ comprising all weakly Cauchy sequences. Is this subspace closed? It is not as immediate as in the case of ...
A sequence $(x_n)$ in a Banach space $X$ is called weakly Cauchy if for every $\ell \in X'$ the sequence $(\ell(x_n))$ is Cauchy in the scalar field. I want to show that weakly Cauchy sequences ...