TITLE: $d(x_n,y_n)$ converges to a limit when $x_n, y_n$ are Cauchy sequences
QUESTION [9 upvotes]: Let $(X,d)$ be a metric space and $x_n, y_n$ Cauchy sequences.  Is there a way to prove that $\lim\limits_{n \to \infty} d(x_n,y_n)$ exists without involving the completion of $X$?  Intuitively you have the $x_n$s, and the $y_n$s, each trapped in an open ball, so their distance cannot change wildly.  But it seems surprisingly difficult to prove that the given limit actually exists.

REPLY [6 votes]: $$d(x_n, y_n) \leq d(x_n, x_m) + d(x_m, y_n) \leq d(x_n, x_m) + d(x_m, y_m) + d(y_m, y_n)$$
so 
$$d(x_n, y_n) - d(x_m, y_m) \leq d(x_n, x_m)  + d(y_m, y_n)$$
Similarly, we have 
$$d(x_m, y_m) - d(x_n, y_n) \leq d(x_n, x_m)  + d(y_m, y_n)$$
so 
$$|d(x_m, y_m) - d(x_n, y_n)| \leq d(x_n, x_m)  + d(y_m, y_n)$$
so $a_n = d(x_n,y_n)$ is a Cauchy sequence in $\mathbb{R}$, so its limit exsits