Let {fn(x)} be defined by: fn(x) = Vx for 0 < x< 1. [a] Prove with an e – N argument that {fn(x)} converges pointwise to the function 1 ;if 0 < x < 1 0 ;if x = 0 f(x) = [b] Show that {fn(x)} does not converge uniformly to f(x) by showing that given any n > 0 you can find a point 0 < x < 1 such that | VI – 1| 2. **Note**: The point æ can be a function of n. [c] Using the e – N definition of uniform convergence, prove that {fn(x)} con- verges uniformly to: f(x) = 1, on the interval [}, 1).
Let {fn(x)} be defined by: fn(x) = Vx for 0 < x< 1. [a] Prove with an e – N argument that {fn(x)} converges pointwise to the function 1 ;if 0 < x < 1 0 ;if x = 0 f(x) = [b] Show that {fn(x)} does not converge uniformly to f(x) by showing that given any n > 0 you can find a point 0 < x < 1 such that | VI – 1| 2. **Note**: The point æ can be a function of n. [c] Using the e – N definition of uniform convergence, prove that {fn(x)} con- verges uniformly to: f(x) = 1, on the interval [}, 1).
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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COURSE: Mathematical/
TOPIC: Uniform Convergence
![Let {fn(x)} be defined by:
fn (x) = Vx for 0 <x < 1.
[a] Prove with an e - N argument that {fn(x)} converges pointwise to the
function
f(æ) = {
1 if 0 < x < 1
0 ;if x = 0
[b] Show that {fn(x)} does not converge uniformly to f(x) by showing that
given any n > 0 you can find a point 0 < x < 1 such that | V – 1| 2.
**Note**: The point x can be a function of n.
[c] Using the e – N definition of uniform convergence, prove that {fn(x)} con-
verges uniformly to: f(x) = 1, on the interval [}, 1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbaa9c753-6b68-41b4-8837-61aa8aa0a19b%2F63922e3f-58ea-4ec4-982f-ad22ff8295d1%2Fujrzu2p_processed.png&w=3840&q=75)
Transcribed Image Text:Let {fn(x)} be defined by:
fn (x) = Vx for 0 <x < 1.
[a] Prove with an e - N argument that {fn(x)} converges pointwise to the
function
f(æ) = {
1 if 0 < x < 1
0 ;if x = 0
[b] Show that {fn(x)} does not converge uniformly to f(x) by showing that
given any n > 0 you can find a point 0 < x < 1 such that | V – 1| 2.
**Note**: The point x can be a function of n.
[c] Using the e – N definition of uniform convergence, prove that {fn(x)} con-
verges uniformly to: f(x) = 1, on the interval [}, 1].
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