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 $f\in X'$.)
