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Let fn E D[0, 1]. Suppose that fh g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show that n00 is Cauchy.

Let fn E D[0, 1]. Suppose that fh g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show that n00 is Cauchy.

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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Let fn E D[0, 1]. Suppose that fh3 g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show that n00 ( fa(=)) neN is Cauchy. Hint: use the Lagrange MVT for F(x) = fn(x) – fm(x) Remark: this implies the existence of pointwise limit f(x) = lim fn(x) and by the Differentiation Limit Theorem, n-00 f'(2) = g(x).
Transcribed Image Text:Let fn E D[0, 1]. Suppose that fh3 g on [0, 1], and lim fn(1/2) exists. For each x E [0, 1], show that n00 ( fa(=)) neN is Cauchy. Hint: use the Lagrange MVT for F(x) = fn(x) – fm(x) Remark: this implies the existence of pointwise limit f(x) = lim fn(x) and by the Differentiation Limit Theorem, n-00 f'(2) = g(x).
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