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1. Let fn (0, ∞) → R where fn(x) (0, ∞)→ R where f(x) = 0. = nx for all n E N and let f (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on (0, ∞). (c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer.

1. Let fn (0, ∞) → R where fn(x) (0, ∞)→ R where f(x) = 0. = nx for all n E N and let f (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on (0, ∞). (c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Real Analysis II Kindly find sample for guide in photo 2. Please solve with hand on paper as it’d easier to understand
1. Let fn (0, ∞) → R where fn(x) (0, ∞) → R where f(x) = 0. = nx for all ne N and let f (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on (0, ∞). (c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer. (d) Does fn converge uniformly to f on [1, ∞)? Prove your answer.
Transcribed Image Text:1. Let fn (0, ∞) → R where fn(x) (0, ∞) → R where f(x) = 0. = nx for all ne N and let f (a) Graph f1, f2, f3, and f on the same axes. (b) Prove that fn converges pointwise to f on (0, ∞). (c) Does fn converge uniformly to f on (0, ∞o)? Prove your answer. (d) Does fn converge uniformly to f on [1, ∞)? Prove your answer.
(0,0) Ex. Let S = [0, 1] x fn(x) = x^ *nexo & 0≤x²1 or 5 to f(x) = { 1 + x = 1 Jaim: {fr} Conv. ptwise (₁,1) f(x)=x² = x to ¼/a ft = (1/2) fa (x) = x² fa= 1/4 f3 (x)=x² 9 f¶ (x) = . =X lim So txES, 200 limm Eanx" lim Bf Suppose 0≤x²1 - Then apo fn(x) = (im X"=0 Cim lim 1^ == Suppose X=1. Then 2-01-1 4 sao fn(x) = fn (x) exists, S₁ = 90 x° S₂ = 96x° +9₁x = 90 +9₁ x S3 = ax +9₁ x + 9₂x² = 90 +3 = ½ Does {fr} Conv. pluise on s? nf Show of = 1 0 Sofn (x) dx = Area of A = 1/2 ² 1/0 +0=1 but So f(x) dx=0 lim S = [0, 1] for 122, define fn (x) = ['n³x if 0≤x≤/n 2-n²(x - ²) 2 % ≤ x ≤ -0 i 3/ / LX ≤I fn 15-01- din f Lim So fm (x) dx # Sr f(x) dx
Transcribed Image Text:(0,0) Ex. Let S = [0, 1] x fn(x) = x^ *nexo & 0≤x²1 or 5 to f(x) = { 1 + x = 1 Jaim: {fr} Conv. ptwise (₁,1) f(x)=x² = x to ¼/a ft = (1/2) fa (x) = x² fa= 1/4 f3 (x)=x² 9 f¶ (x) = . =X lim So txES, 200 limm Eanx" lim Bf Suppose 0≤x²1 - Then apo fn(x) = (im X"=0 Cim lim 1^ == Suppose X=1. Then 2-01-1 4 sao fn(x) = fn (x) exists, S₁ = 90 x° S₂ = 96x° +9₁x = 90 +9₁ x S3 = ax +9₁ x + 9₂x² = 90 +3 = ½ Does {fr} Conv. pluise on s? nf Show of = 1 0 Sofn (x) dx = Area of A = 1/2 ² 1/0 +0=1 but So f(x) dx=0 lim S = [0, 1] for 122, define fn (x) = ['n³x if 0≤x≤/n 2-n²(x - ²) 2 % ≤ x ≤ -0 i 3/ / LX ≤I fn 15-01- din f Lim So fm (x) dx # Sr f(x) dx
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