TITLE: Show that if $\{x_{n}\}$ and $\{y_{n}\}$ are Cauchy sequences in X then $d(x_{n},y_{n})$ converges in $\mathbb{R}$.
QUESTION [8 upvotes]: Could someone help me through this problem?
Let $X$ be a metric space. Show that if $\{x_{n}\}$ and $\{y_{n}\}$ are Cauchy sequences in $X$ then $d(x_{n},y_{n})$ converges in $\mathbb{R}$.
Does this follow from the fact that every Cauchy sequence in $\mathbb{R}$ is convergent?

REPLY [6 votes]: This is essentially Davide Giraudo's approach, but somewhat shorter: The triangle inequality gives
$$d(x_m,y_m)\leq d(x_m,x_n)+d(x_n,y_n)+d(y_n,y_m)$$
or
$$d(x_m,y_m)-d(x_n,y_n)\leq d(x_m,x_n)+d(y_n,y_m)\ .$$
As the right side is symmetric in $m$ and $n$ we have in fact
$$\bigl|d(x_m,y_m)-d(x_n,y_n)\bigr|\leq d(x_m,x_n)+d(y_n,y_m)\ .$$