"Suppose (a_n) is a bounded increasing seq and (b_n) is a bounded decreasing seq. Let x_n=a_n+b_n. Show that $\displaystyle \sum|x_n-x_{n+1}|$ converges."

Triangle ineq yields

$\displaystyle \sum|x_n-x_{n+1}|\leq \sum|a_n-a_{n+1}| + \sum|b_n-b_{n+1}|$

from which point I am unable to deduce that either of the RHS sums converge.