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Q: 1-Let B be the set of all infinite sequences over {0, 1}. Is this set countable or not, prove it?

A: Please post the multiple parts separately.Here I answered the 1st question as per our policy.

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Q: . Let {n} and {n} be Cauchy sequences in R. Prove that {rnyn} is also a Cauchy se- quence in R.

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Q: II. Prove if X and Y are the sequence of real number and converges to x a real number then product…

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A: Consider the provided question, We have to check the given statement is true or false, Part first:

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A: please see the next step for solution

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Q: Let {an} and {bn} be two Cauchy sequences of real numbers. Then show that the sequence {|an − bn|}…

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Q: is it kue that if Can) and Cbn) two sequences of real humbers such Hhat : ansbu for all -nE N i aud…

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Q: is it seies converge ost not. 1-2)^

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Q: Let (an) be a bounded sequence. Define (b,) by bn := sup{an, an+1, An+2;...}. Prove that (bn)…

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A: True. I have proved using definition of cauchy and convergent sequence.

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A: Given: to prove, limn→∞an+an+1+.....an+r=0,r≥0

Q: ODoes an In(n) – In(n + 4) converge or diverge?

A: For convergence test, apply limn→∞ on an

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Q: Every Cauchy sequence in the Euclidean metric space R" with n a positive integer is convergent. O…

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Q: be sequence in R, dis discrete %3D then (.) is not Cauchy sequence O True O Fabe

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Q: 2- Show that the Sequence An define by and dn=Vant 2 for 'nI %3D is a monotonic increasing. a

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Q: Prove that ∑^infinity_(n=1) ((-1)^n/n) converges.

A: Apply Ditichlet theorem.

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Q: Pir Prove that it (bn) is boundel sequence of read numbers. and limpz(an )=0, then limna( bmen)

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Q: 1. Show that the sequence fh3, whore fnex) = tan nx , x7o is uniprmly on ony Interval Ca,6], a> o…

A: Note: Our guidelines we are supposed to answer only one question. Kindly repost other question as…

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A: The given question can be solved as shown in step2.

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