4. ) Suppose that (an) is a sequence of nonnegative numbers such that lim an = 0. Prove that there is some subsequence (an) of (an) such that n-00 Lk=1 anx Converges. Hint: Use the fact that an → 0 to show you can find a subsequence satisfying an <2-k for all k.

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A: we have to use definition of limit of sequence.

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A: We will use Triangle inequality.

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A: To prove limn→0(-1)nnn2+1=0

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A: The given question is not well defined but still we have given a proof.   See the attachment

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Q: Assume that the sequence shown below converges and find its limit. √3, √3+√3,√√3+√3+√3 √3+₁ " 3 +…

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Q: 3. Let (Sn) be a sequence of nonnegative real numbers, and suppose lim s, = 0. Prove that lim S, =…

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Q: 81. Happy Thanksgiving!!

A: We have to deduce that an is convergent and find its limit.

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Q: (d) (i) Define: sequence of real numbers {x} converges to the limit x = R. 3n+1 7n-4 (ii) Use the…

A: d.1. Definition:          A sequence of real numbers xn is said to be converges to the limit x∈ℝ…

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A: Step 1:Step 2: Step 3: Step 4:

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