Convergent Telescoping Sum with Bounded Sequences (a_n) and (b_n)

[Treadstone 71]

"Let (a_n) be bounded decreasing and (b_n) be bounded increasing sequences. Let x_n =a_n+b_n. Show that $$\sum|x_n-x_{n+1}|$$ converges."

This ALMOST is a telescoping sum, but it doesn't work since if I try to use the triangle inequality, the sum I want is on the greater side. Ratio test, root test, etc all fail since there is insufficient information.

 

[shmoe]

How did you try to apply the triangle inequality? Try different groupings of the terms.

 

[benorin]

By the triangle inequality,

$$\sum|x_n-x_{n+1}| = \sum|a_n+b_n-(a_{n+1}+b_{n+1})| \leq \sum|a_n-a_{n+1}|+\sum|b_n-b_{n+1}|$$​

also, bounded monotonic sequences are convergent.

 

[Treadstone 71]

They are indeed convergent, but showing that the terms of the series go to 0 is insufficient to show that it converges. If we remove the absolute value signs, it becomes a telescoping sum and therefore it converges. Knowing that, and that an, bn converge, can we conclude something from it?

 

[shmoe]

They are indeed convergent, but showing that the terms of the series go to 0 is insufficient to show that it converges. If we remove the absolute value signs, it becomes a telescoping sum and therefore it converges. Knowing that, and that an, bn converge, can we conclude something from it?

Yes, you will have shown the right hand side converges to a finite value. The left hand side has positive terms and is then bounded above, so?

 

[Treadstone 71]

How did you conclude that the RHS converges?

 

[Treadstone 71]

Nevermind, I figured it out, thanks.