• Exercise 2 Assuming that i) the sequence of functions {fn(x), x E [0, 2]} converges to f(x) in L2[0, 2]. ii) the sequence of functions {In(x), x E [0, 2]}, converges to g(x) in L2[0, 2], prove that lim xp (x)“6(x)"f f(x)g(x) dx. You can only use without proof that: i) The Cauchy-Schwarz inequality holds. ii) A convergent sequence is bounded.
• Exercise 2 Assuming that i) the sequence of functions {fn(x), x E [0, 2]} converges to f(x) in L2[0, 2]. ii) the sequence of functions {In(x), x E [0, 2]}, converges to g(x) in L2[0, 2], prove that lim xp (x)“6(x)"f f(x)g(x) dx. You can only use without proof that: i) The Cauchy-Schwarz inequality holds. ii) A convergent sequence is bounded.
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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Homework exercise.
![Exercise 2 Assuming that
i) the sequence of functions {fn(x), x € [0,2]} converges to f(x) in Lə[0, 2].
ii) the sequence of functions {gn(x), x E [0, 2]}, converges to g(x) in L2[0, 2], prove that
r2
fn(x)gn(x) dz = [ f(x)g(x) dxr.
lim
0.
You can only use without proof that:
i) The Cauchy-Schwarz inequality holds.
ii) A convergent sequence is bounded.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44d4bb1e-783f-43e8-9cab-fd5fb3199e14%2F2aa91ea3-80e3-4adf-8732-dc1507c2fd53%2F9jqidl8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 2 Assuming that
i) the sequence of functions {fn(x), x € [0,2]} converges to f(x) in Lə[0, 2].
ii) the sequence of functions {gn(x), x E [0, 2]}, converges to g(x) in L2[0, 2], prove that
r2
fn(x)gn(x) dz = [ f(x)g(x) dxr.
lim
0.
You can only use without proof that:
i) The Cauchy-Schwarz inequality holds.
ii) A convergent sequence is bounded.
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