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

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A J
A J 2016-2-10
评论： Walter Roberson 2016-2-12
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?

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Torsten
Torsten 2016-2-10
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.
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