Answered: Let 1 ≤ p < ∞ and let {fn} be a…

Let 1 ≤ p < ∞ and let {fn} be a sequence of functions in Lp[a, b]. Suppose there is a function f ∈ Lp[a, b] with limn→∞ ||fn − f||p = 0 . Prove that {fn} is a Cauchy sequence in Lp[a, b].

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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Question

Let 1 ≤ p < ∞ and let {fn} be a sequence of functions in
Lp[a, b]. Suppose there is a function f ∈ Lp[a, b] with
limn→∞ ||fn − f||p = 0 .
Prove that {fn} is a Cauchy sequence in Lp[a, b].

(16) Let 1 <p<00 and let {Sn} be a sequence of functions in
LP la, 6). Suppose there is a function fe LPla, b] with
lim If.- flp 0.
Prove that {fa} is a Cauchy sequence in LP(a, b).

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