= Solution
For fixed $m$, the <reverse triangle inequality> gives
$$
|d(x_m,x_n)-d(x_m,x_k)|\leq d(x_n,x_k).
$$
The right side tends uniformly to zero as $n,k\to\infty$ because $(x_n)$ is a <Cauchy sequence>. Hence $(d(x_m,x_n))_n$ is a real <Cauchy sequence>. By <completeness> of $\mathbb R$, it has a finite, nonnegative limit:
$$
\boxed{d_m=\lim_{n\to\infty}d(x_m,x_n)\geq0.}
$$
This conclusion uses completeness only of the real line, without assuming that $X$ is complete.
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