Does the sum of bounded increasing and bounded decreasing sequence is convergent?

Question

Let $(a_n)$ be a bounded decreasing sequence and $(b_n)$ be a bounded increasing sequence and if $x_n = a_n +b_n. $ Claim $ \sum_{n =1 }^{\infty } | x_n - x_{n+1} |$ is convergent.

My Claim :

$| x_n - x_{n+1} | $ is convergent if and only if sequence of partial sum is bounded.

Let $A =$ Inf $\{a_n \}$ and $B =$ Sup $\{b_n \}$

Our required series is lower bounded by $A +b_1$ and upper bounded by $B +a_1.$ Does that imply that $ \sum_{n =1 }^{\infty } | x_n - x_{n+1} |$ is convergent.

Answer 1

Hint:

$$|x_n - x_{n+1}| = |a_n + b_n - a_{n+1} - b_{n+1}| \le |a_n - a_{n+1}| + | b_n - b_{n+1}|,$$

so it suffices to show that both

$$\sum |a_n - a_{n+1}|, \sum |b_n - b_{n+1}|$$

are bounded. These two are much easier as they are monotone and bounded