Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a...

Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Expert Solution

Solution:

Given That Let {an}n∈N be a sequence with lim n→+∞ an = 0. Prove that there exists a subsequence {ank }k∈N so that X∞ k=1 |ank | ≤ 8

Poof:

We have show that for each k we can choose an Ank so that |Ank| <1/2k.

Then the comparison test shows thatΣAnk is absolutely convergent, hence convergent.

Startwith k = 0. If there were no An0 with absolute value < 1,then 1 would be a lower bound for the whole of {|An|},contradicting the assumption that lim inf |An| = 0. Choose n0so that |An0| < 1.

Now there must be an n1 > n0 suchthat |An1| < 1/2, or else 1/2 would be a lower bound for{|An|}n > n0, and so lim inf |An| would be at least1/2, a contradiction.

Choose An1, and now, arguing the sameway, we can find n2 > n1 with |An2| < 1/4. Proceedingthis way, for every k we find Ank with |Ank| < 1/2k.

Colby Messinger answered 1 year ago

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