3. Suppose {fa) is an equicontinuous sequence of functions on a compact set K, and U.) converges pointwise on K. Show that (fn converges uniformly

Q: |an also converges because 0 If an converges then < lan|, for n = 1, 2, ... n=1 n=1

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Q: Consider the sequence n²x(1 – nx) x € [0, 1/n] { - fn(x) := elsewhere . Does it converge uniformly…

A:

Q: {xn} is a bounded sequence such that all convergent subsequences converge to the same number

A: Every bounded sequence has convergent subsequence.

Q: 00 Show that converges if [r| < 1. п: п

A: The given series is

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

A: Proof

Q: Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1

A:

Q: 7. a) Let x = c, with ||- Give an example of a Cauchy sequence in X that do not converge in X.…

A: Here, X=C00  is set of all sequence having only finitely many non zero terms with the norm xp where…

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

A: Recall.   1: Triangle inequality for integrals.   2: if (a_n)^2 goes to 0 as n goes to infinity then…

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: 7. Suppose A and B are compact subsets of R and that fn is a sequence of functions converging…

A:

Q: f,(z) = 1+ 1+ nz

A:

Q: Let f(x) = (ncosx)/(n+ex) x in [0,1], n in N Prove that the sequence converges uniformly on [0,1].

A:

Q: 2. a. Prove using the formal definition of a limit that the following sequence converges: -n3 + 2n –…

A: This is a problem of sequence.

Q: Assume (fn) and (gn) are uniformly convergent sequences of functions. (b) Give an example to show…

A: Given : Assume (fn) and (gn) are uniformly convergent sequence of functions.

Q: Show that > iP converges to a finite value iff p < –1. i=1

A:

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…

A:

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

A:

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

A:

Q: Let {fn(x)} : „x ER and > 0. 1+ nx Show that ift>0, the sequence {fn(x)} converges uniformly on

A: Given: To explain the given statement as follows, To show that if t&gt;0, the sequence fnx converges…

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

A:

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

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Q: an Show that if lim =0 and E-i b, converges then En=1 converges too. an =1 n+ 00

A: Given that  limn→∞anbn=0  and  ∑n=1∞bn converges.

Q: (b) Give an example to show that the product (fngn) may not converge uniformly.

A:

Q: Use the Limit Comparison Test to determine whether converges or diverges. Vnvn +1 n=1

A: we have to test whether the given series is convergent or divergent by limit comparison test

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: ) Prove that if XnXnand YnYn are any two sequences of real numbers such that XnXn converges to xx…

A:

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

A:

Q: Let {x„}" and {ym}*=1 be convergent sequences with limits x and y, respectively. Suppose that all…

A:

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

A:

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

A:

Q: What are all of the values of x, such that converges?

A: We have to use GP sum Common Ratio.

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: 3- a- By definition show that (x,) = (+3) converges to 3. b- Does (+3) bounded, Cauchy? Show by part…

A:

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

A:

Q: Show that the sequence of functions {f.} where f,(x)=nxe is non-uniformly convergent on [0,1]. (by…

A: Test for uniform convergence of sequences: Let fn be a sequence of functions such that it converges…

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

A:

Q: Let f : A → R and (xn),(Yn) are two sequences converges to c such that Xn, Yn # c, Vn E N. If…

A: We understand this statment by the help of an counterexample .

Q: Let (rn) be a sequence of real numbers. Show that the convergence of (sn) implies theconvergence of…

A: We have to prove that if convergence of (sn) implies the convergence of (|sn|).

Q: Let fn(x) x +n b. Use the Lemma for non-uniform convergence to show that the sequence does not…

A:

Q: 14. Suppose (an) is a bounded sequence such that all of its converging subsequences converge to the…

A:

Q: Suppose {fn}n=1 and {gn}n=1 converges uniformly on E where {fn}n=1 and {gn}n=1 are sequences of…

A:

Q: Exercise 9. Let {an}_1 be a bounded sequence and {b„}=1 be a sequence converges to 0. Prove that the…

A: Given that an is bounded. Then there exists M&gt;0, such that an≤M for all n. Thus we get  anbn≤Mbn…

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

A:

Q: an Show that if lim bn =0 and =1 n- 00 converges then En=1ªn converges too. n=D1

A: We have to solve given problem:

Q: 8. Suppose ar > 0 and Does > ak converge or k diverge? Why? %3D

A:

Q: (a) Suppose that f : X → Y is a continuous function and {xn} is a Cauchy sequence in X. Does {…

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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…

A:

Q: 1. Show that the sequence ((nx - - 1/n)e n) converges uniformly on [0, 0). (15 points)

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

A: