Suppose {f,} is a sequence in Banach space B s.t. |[fn - fn=i|l s /2-). Show that the sequence {f,} is convergent in B.

Q: Prove that if the sequence {an} is divergent and the sequence {bn} is convergent, then the sequence…

A: Given: The objective is to prove that if the sequence {an} is divergent and the seqence {bn} is…

Q: Q4] Show that the sequence (xn) = (T") for all n EN is not a Cauchy .sequence

A: A sequence {Xn}of real numbers is called a Cauchy sequence if for every positive real number ε,…

Q: 3) The space C is the set of all Convergent sequences in e. please Prove that C is complete.

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Q: Let X be a Banach space an {xn} be a sequence in. X. Prove that if {xn} converges in norm in X, then…

A: Given: Suppose X is a Banach space and xn→x in X. To prove: xn→x weakly

Q: Let f be a uniformly continuous function on (a, b) and suppose that {xn} is a Cauchy sequence in…

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Q: 3. If a sequence (rn) in a metric space X is convergent and has limit r, show that every subsequence…

A: We have a sequence ( xn ) in a metric space X which is convergent with a limit x. Then, we have to…

Q: Prove that a sequence in ℝ can have at most one limit

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Q: Suppose a Cauchy sequence {xn} is such that for every M ∈ N, there exists a k ≥ M and an n ≥ M such…

A: every cauchy sequence of real numbers is convergent. And a sequence can have at most one limit.

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: 3. Prove that a sequence {a,} converges to the real number a if and only if lim sup a lim inf an =…

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Q: 5. Let {fn} be a sequence of measurable functions on E = [0, 1] satisfying Prove that the sequence…

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Q: Suppose that (an)n20 is a bounded sequence of real numbers. For each natural number n, define fn(x)…

A: We are given a sequence of functions, where {an} is a bounded sequence. We need to establish the…

Q: Prove that if a sequence (an)1 is not bounded, then there is a subsequence (akn)-1 which either…

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Q: 4. Let f: DCR→ R be uniformly continuous and {xn} be a Cauchy sequence in D. Then show that {f(Xn)}…

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Q: Suppose that the sequence of functions {fn} and the function f are integrable on [a, b] and that…

A: Given that,  The functions fn and f are integrable on a,b. fn converges to f on uniformly a,b. an…

Q: Let (fn) be a sequence in L that converges uniformly on 2 to f. If µ(2) < ∞ show that…

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Q: If fn is a uniformly equicontinuous sequence of functions on a compact interval and fn → f…

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Q: A Cauchy sequence f} in L₂ converges to an element f in L₂

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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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Q: a. In a metric Space (X, d) if (xn) and (yn) are two sequences in X such that : ХX, — хеХапd yn —…

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Q: 2. Show that if a > 0, then the sequence (n²a?e¬n*) coverges uniformly on the interval [a, ∞), but…

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Q: 6. Let {fn} be a sequence of measurable functions on E = (0, 1] satisfying %3D Prove that the…

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Q: Suppose Sanx",Σ bnan converges for all x in (-1,1). Σana" = Σbnan in a subset E of (-1,1) implies…

A: It is given that,  ∑anxn and ∑bnxn are convergent for all x in (-1, 1). Then,  ∑anxn = ∑bnxn in a…

Q: Prove that the sequence {n} defined by Xn = 1+ is a Cauchy sequence.. ¹ + 1 + 1 + 4 9 +

A: Introduction: A sequence is called a Cauchy sequence when its terms eventually become arbitrarily…

Q: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

A: Show that the sequence of functions fn(x), where  fnx=x1+nx2,  x∈ℝ is non-uniformly convergent on…

Q: Suppose that (an) and (bn) are convergent sequences of real numbers with limits an → a and br → b,…

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Q: 4. Show that the sequence {fn}, where fn(x) = xER 1+n°x*' is point-wise convergent but is not…

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Q: Suppose {f,} is a sequence in Banach space B s.t. ||fn - fn+1|| s'/2-). Show that the sequence {f,}…

A: For a vector space X, over a field F, a normed linear space X,· is referred to as Banach space if…

Q: If H be a Hilbert space and C be a subset of H. Give an example of a function, T:C- C, which…

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Q: 4. Let f : DCR → R be uniformly continuous and {In} be a Cauchy sequence in D. Then show that {f(1)}…

A: Image of a Cauchy sequence

Q: (a) If (xn) is a Cauchy sequence in the discrete metric space (M,d), then prove that for some NEZ*,…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

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: 3. If g:A → R is continuous and (an) is a Cauchy sequence in A, does it follow that the sequence…

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Q: 4. Let f : DCR → R be uniformly continuous and {rn} be a Cauchy sequence in D. Then show that…

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Q: 2.21, Suppose that {an}n€N is a Cauchy sequence in a metric space X, d suppose there exists a…

A: We need to show xn converges to x

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

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Q: Suppose (an) is a sequence of positive real numbers such that an+1 Prove that liman (an) even…

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Q: Let {an}=1 be a convergent sequence such that a1 = V5 and an+1 V5 an. Compute lim an.

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Q: If {xn} is a bounded sequence on R. Prove: lim n→+∞

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Q: Find the limit of the sequence {an}, or show that it diverges: An 2n +1 In

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Q: Consider the sequence S n²x(1 – nx) { x € [0, 1/n] fn(x) := elsewhere. Investigate the pointwise…

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Q: State the definition of a sequence xn converting to a point x in a metric space (X, d). prove that…

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Q: Prove that any convergent sequence {xn}=1 is bounded.

A: We want to prove every convergent sequence is bounded.

Q: ** Consider the sequence (xn) in R, where n (-1)". Show that (rn) doesn't %3D converge to any limit.

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Q: Suppose {a,} is monotonic. Prove that {an} converges if and only if {a} converges

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Q: 4. Show that the sequence { fn}, where nx fa(x) = xER 1+n°x2 is point-wise convergent but is not…

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Q: 4. Let f : DCR R be uniformly continuous and {r,} be a Cauchy sequence in D. Then show that {/(xn)}…

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Q: 4. Show that the sequence {fn}, where nx fn(x)=- xER 1+n°x' is point-wise convergent but is not…

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Q: 4. Show that the sequence (n) does not converge uniformly on [0, 2] by showing that the limit…

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