a) Let (h,). (t) be sequences of hounded functions on A that converges uniformly on A to h,t respectively. Show that (h,tn) converges uniformly on A to ht.

Q: Q2 Prove that the scries 2+ x2 3+X2 4+x is Unitormly convergem? in any interval?

A: The given series is: 11+x2+12+x2+13+x2+……. Therefore the given series can be written as:…

Q: Prove that if r >1 then f r" dx converges to 1

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Q: Exercise 3. Find f(x) = lim,0 fn(x) on S. Show that (fn)n converges uniformly to f on closed subsets…

A: Answer

Q: Let (h,), (tn) be sequences of bounded functions on A that converges uniformly on A to h, t…

A: Given that there are two sequence (hn) and (tn) be sequence of bounded function on A that converges…

Q: Suppose fn:A→ Ris uniformly continuous on A, Vn E N and fn →f converges uniformly on A. Question:…

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Q: Suppose that |f(r)|I b where and g(r)dx converges. Prove |f(x)dr converges. that

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Q: Let g : [0, 1] → R be continuous and g(1) = 0. Define gn(x) = g(x)x" for x E [0, 1] and n = 1,2,…

A: As per our guidelines we are supposed to answer only one question.

Q: Theorem 64. Let {pk}1 converge to point If y # x, then {pk}1 does not converge to y. x.

A: Given that sequence pkk=1∞ converges to point x.

Q: 8. Let f : R → R be uniformly continuous on R and if fn(x) = f(x + ) for x E R, show that (fn)…

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Q: If |fn(x)| < an for all r and E=1 an converges, prove that En=1 fn(1) converges uniformly.

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Q: Let fn (x) uniformly converge to f(x) on [a, b]. If fn(x) E R[a, b], then f(x) E R[a, b] True False

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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: ounded funclions on A to h,t converges uniformly

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Q: Does the Bounded Convergence Theorem hold if m(E)<∞ but we drop the assumption that the sequence…

A: Given below clear explanation

Q: Show that for every positive function f on R there is a monotone increasing sequence of integrable…

A: We need to show that for every positive function f on ℝ there is a monotoneincreasing sequence of…

Q: 2.1 Show that a sequence of Riemann-Integrable functions fn(r) converges uniformly to a…

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

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Q: Let fn(x) = x¹(1-x), r [0, 1] and n € N. Show that fn = 0 on [0, 1] but that {f} is not uniformly…

A: Given fn(x) =xn(1-x),             x∈[0,1] and n∈N To prove: {f'n} is not uniformly convergent.

Q: If 2 a, converges, then la,| diverges. n=1 n=1 True False

A: Given statement is FALSE. I HAVE given a counter -example.

Q: (c) Show that fn (x) : x2+n? converges uniformly on [0, 0).

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Q: Exercise 2. Determine whether the sequence of functions (fn)n converges uniformly on S. fn(x) = nx"…

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Q: Show that if each fn is bounded on A and (fn) converges uniformly to f on A, then f is bounded on A.

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Q: Let (fn) be a sequence of functions that are integrable on [a, b] and that converge uniformly on [a,…

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Q: If f(x) ≤ g(x) for all x in [a, +∞o) and converges. •+∞ f(x) da converges, then [*** g(x) dx also

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Q: Two sequences of functions fn and gn that converge uniformly but fn In does not converge uniformly.

A: See the attachment

Q: Compute the point-wise limit of (fn) and determine if they converge uniformly on given intervals. 1.…

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Q: Let (hn),(tn) be sequences of bounded functions on A that converges uniformly on A to h,t…

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Q: 4. Suppose fn:A→ Ris uniformly continuous on A , Vn e N and fn →f converges uniformly on A.…

A: Given: fn: A→ℝ is uniformly continuous on A ∀n∈ℕ To prove: fn→f converges uniformly on A  fn→f…

Q: the sequence {fn} converges uniformly to f on R. Prove that the limit function - R be bounded.…

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Q: If the interval of convergence for f(x) = E anx" is [-R, R], then an x" converges uniformly on [–R,…

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Q: Let fn(x) x + n a. Show that fn(x) converges pointwise [0, 0). on

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Q: 5. Let h, : [0, o∞) → R be given by h„(x) and let h : [0, ∞) → R be given by I +n %3D h(x) = = x.…

A: Answer

Q: Let (Jn)neN be a sequence of functions f: R → R satisfying |fn+1(c) – fn(c)| 2 and neN , and ae(0,…

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Q: A sequence {ann21 contained in the interval [0, 1] is uniformly distributede if and only if lim -- […

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Q: Prove with an e – N argument that {fn(x) + gn(x)} converges uniformly to f (x) + g(x) on R.

A: The objective here is to prove that : fnx+gnx converges uniformly to fx+gx Given: fnxconverges…

Q: Prove that if a sequence of continuous functions fn :R→R is uniformly convergent on Q, then it is…

A: Given function fn:ℝ→ℝ sequence of continuous function. fn is uniformly convergent on ℚ To prove: fn…

Q: Let h: [0,1] R be a continuous function with h(1) by fn(x) = x"h(x) for x e [0,1]. Show that {fn}…

A: A sequence of real-valued functions considers two types of convergence. These are pointwise…

Q: Let fn(x) = nxe-n, for x = [0, 1]. (a) Show that for each x € [0, 1] we have fn(x) converges to 0.…

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Q: Let fn(x) = x¹(1+x¹)−¹. Use the uniform norm to determine if the sequence (fn) uniformly converges…

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Q: Let (hn), (tn) be sequences of bounded functions on A that converges uni formly Sh, tn) converges…

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Q: Let fn E C[0, 1]. Suppose that fn (x) converge uniformly on [0, 1]. 1. Show that (fn) is uniformly…

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Q: Exercise 5. Consider the sequence (fn)n defined by fn(x) = 1#nz: for r>0. -.... 3. Show that (fn)n…

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Q: 3. Let {fn}n=1 and {gn}-1 converges uniformly on E. Prove that {fn - In}n=1 converge uniformly on E.

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Q: Is the following statement correct: If (fn) is a sequence of differentiable functions on [a, b] such…

A: Given statement is

Q: 5. Let (n) be a uniformly bounded sequence of functions which are Riemann integrable on [a, b, and…

A: By the hypothesis there exists an real number M such that Supremum of x belongs to [a,b] of…

Q: The set S on wc fn Cx) = 1+ x" Converges pointwire ?

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Q: If |fn(x)| < an for all x and Ln=1 an converges, prove tha En-1 fn(x) converges uniformly. n=1

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Q: Consider the sequence of functions {fn}n=2 where fn: [a, b] → R defined for n > 2 by if 0 sxs fa =…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.