suppose that an is an increasing sequence and bn is a decreasing sequence with lim (bn - an) = 0. Prove that lim an and lim bn both exist and lim an = lim bn

Hi i am given this as homework and i am not sure how to start. I think i know how to prove the second part, but how do i show that lim an and lim bn exists?

[Walter Roberson]

This was not a MATLAB question and should not have been posted to MATLAB Answers.

Answer 1

[Torsten]

Hint:

an = bn - (bn-an) <= b1 - (bn-an)
bn = an + (bn-an) >= a1 + (bn-an)

and

bn-an -> 0

Thus an is bounded from above and bn is bounded from below.

What do you know about increasing sequences bounded from above and decreasing sequences bounded from below ?

Best wishes

Torsten.

[A J]

bounded monotone sequences are convergent! So since they are bounded and increasing/decreasing, they are convergent -> therefore their limits exist?? By the monotone convergence theorem?

[Torsten]

Yes.

Best wishes

Torsten.

[A J]

thank you so much!