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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 is clear from the estimate $$|d(y,x_m)-d(y,x_n)|\leqslant d(x_m,x_n)$$ that a Cauchy sequence in $... It seems that the term weekly Cauchy sequence is more often used in a different meaning. (For a Banach space$X$, the sequence$(x_n)$is weakly Cauchy if$f(x_n)$is Cauchy for every$f\in X'\$.)