(3) Let {an} be a sequence in R that has a subsequence {an;} satisfying |ani — ani+1| ≥c for some constant c> 0 for all i Є N. Prove that {an} diverges.

Q: Prove that the sequence {sin n + cos n is bounded. =1

A: The given sequence is We have to prove that the sequence is bounded.

Q: 3. True or False? Prove your answer! a) Suppose the sequence (xn) does not converge to 0. Then, for…

A: A sequence is said to be converge if it has limit (finite)

Q: Let xn be a sequence in a metric space X. Suppose that x, has a subsequence xn, converges to a point…

A: Given, xn  be a sequence  in a metric space X and  xn has a subsequence xnjthat converges to  a…

Q: Let {xn} be a bounded sequence of real numbers. Prove that {xn} converges if and only if lim inf xn…

A: Proof

Q: Prove that if {xn} is a sequence such that 1+ n |æn| < for all n E N, 1+ n2 then the sequence {xn}…

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Q: 1.) Prove that the sequence An= nt n In (2+1) converges to O. That is, prove limln (^+1)=0 nyo

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Q: 1 Let {fn} bes sequence of real nonnegative functions on R', consider the following four statements:…

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Q: Assume you have given a sequence c,, with non-zero values (cn #0 for n = 1,2,...) that fulfils the…

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Q: Let {ai} be a sequence in a metric space X,d. Prove that lim an = p if and only if lim d(an, p) =…

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Q: Is {3n/(3n-1)} n=1 to infinity, a convergent sequence in R with the usual metric dR(x,y) = abs(x-y)?

A: The given sequence is .We need to check that the given sequence is convergent or divergent with the…

Q: 3. Let {an be a sequence in R. (a) Show that it has a monotonic subsequence. (b) Prove that a…

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Q: 14. Prove that every Cauchy sequence is bounded (Theorem 1.4)

A: To prove: Every Cauchy sequence is bounded.

Q: Let x be the sequence X 1, 2, 3, Select ALL of the correct statements (note that there may be more…

A: Note the following properties about sequence and subsequence : Every subsequence of convergent…

Q: 00 Let r be a constant. Suppose ' converges to -1, then the sum of )r' is i-1 i-0

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Q: Let Xn=cos пл 2 then limit supremum of the sequence is

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Q: In(1+2e") d) an n

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: 3. Prove that if (a, } and (b,} are Cauchy sequences of rational numbers satisfying lim (a, - b, = 0…

A: 3.

Q: i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If SnX1 X2 + x3…

A: Given that, { xn } is a sequence of non negative terms and this sequence is monotonic decreasing,…

Q: Suppose that the function f:R → R be differentiable and that {xn}is a strictly increasing bounded…

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Q: 2) Prove that rxz(1) <rzz(0), where r denotes the autocorrelation function of a sequence r(n) and I…

A: Autorrelation function is mathematically given by, rxx(l) = E[X(t+l).X(t)] , for all t. Similarly,…

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

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Q: Show that a, converges to A iff {a, - A) converges to 0.

A: Answer

Q: Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. a)…

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Q: Prove that the function f : R → R is not continuous at xo = f(x) = { 2 +1 r40 if x E Q if x 4 Q…

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Q: 1 then (an) converges to 0. An+1 Exercise 8. Prove that if (xn) is a sequence such that lim n→∞ An…

A: Given that, 8. The sequence is limn→∞an+1an=12 then an converges to 0. 9. The sequence is…