0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.