Find the mistake in the following “proof": "Statement". If fn : X → Y is a sequence of functions converging uniformly on X to a function f, and if f is continuous at xo, then there exists N e N such that fn is continuous at xo for all n > N. "Proof". Let ɛ > 0. By uniform convergence, we can find N e N such that Vn > N, Vx E X : dy (fn(x), f (x)) < By continuity of f at xo, we furthermore find 8 > 0 such that Vx € X : dx(x, xo) < 8 → dy(f(x), f(xo)) < Hence whenever n > N and dx(x,xo) < & we get dy(fn(x), fn(xo)) < dy(fn(x), f(x)) + dy(f(x), f(xo)) + dy (f(xo), fn(x0)) = E. Hence fn is continuous at xo for all n >N.
Find the mistake in the following “proof": "Statement". If fn : X → Y is a sequence of functions converging uniformly on X to a function f, and if f is continuous at xo, then there exists N e N such that fn is continuous at xo for all n > N. "Proof". Let ɛ > 0. By uniform convergence, we can find N e N such that Vn > N, Vx E X : dy (fn(x), f (x)) < By continuity of f at xo, we furthermore find 8 > 0 such that Vx € X : dx(x, xo) < 8 → dy(f(x), f(xo)) < Hence whenever n > N and dx(x,xo) < & we get dy(fn(x), fn(xo)) < dy(fn(x), f(x)) + dy(f(x), f(xo)) + dy (f(xo), fn(x0)) = E. Hence fn is continuous at xo for all n >N.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Uniform converg
Transcribed Image Text:Find the mistake in the following "proof":
"Statement". If fn : X → Y is a sequence of functions converging uniformly on X to a
function f, and if f is continuous at xo, then there exists N E N such that fn is continuous
at xo for all n > N.
"Proof". Let ɛ > 0. By uniform convergence, we can find N EN such that
Vn > N, Væ e X : dy(fn(x), f(x)) <
3
By continuity of f at xo, we furthermore find 8 > 0 such that
Vx E X : dx(x, xo) < 8 → dy(f (x), f(xo)) <
6.
Hence whenever n > N and dx(x, xo) < 8 we get
dy (fn(x), fn(x0)) < dy(fn(x), f(x)) + dy (f(x), f(xo)) + dy (f (x0), fn(x0))
3
3
= E.
Hence fn is continuous at xo for all n >N.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
