If $(x_n)$ and $(y_n)$ are Cauchy sequences in a metric space X with metric d, how would I show the sequence $(d(x_n,y_n))$ converges?
I'm supposed to use $d(x_n,y_n)\leq d(x_n,x_m)+d(x_m,y_m)+d(y_m,y_n)$ If I subtract by $d(x_m,y_m)$ on both sides:
$d(x_n,y_n)-d(x_m,y_m)\leq d(x_n,x_m)+d(y_m,y_n)$
Since $d(x_n,x_m)$ and $d(y_m,y_n)$ are Cauchy, then each can be less than $\frac{\epsilon}{2}$ which makes the left side $\leq \epsilon$. How would I show it converges though?