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Consider the decreasing sequence of functions {f„} defined on R, where fn(x) = X[n,00), Vn e N. That is, fn is the characteristic function of the interval [n, 0) for all n ɛ N. Let ƒ = 0 on R or f(r) = 0 for any re R. Show that fn + f pointwise on R, that is, lim fn(r) = 0, Vr E R.

Consider the decreasing sequence of functions {f„} defined on R, where fn(x) = X[n,00), Vn e N. That is, fn is the characteristic function of the interval [n, 0) for all n ɛ N. Let ƒ = 0 on R or f(r) = 0 for any re R. Show that fn + f pointwise on R, that is, lim fn(r) = 0, Vr E R.

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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Consider the decreasing sequence of functions {f„} defined on R, where fn(x) = X[n,00), Vn e N. That is, f, is the characteristic function of the interval [n, 0) for all n e N. Let f = 0 on R or f(r) = 0 for any r e R. Show that fn + f pointwise on R, that is, lim fn(r) = 0, Vr e R. %3D
Transcribed Image Text:Consider the decreasing sequence of functions {f„} defined on R, where fn(x) = X[n,00), Vn e N. That is, f, is the characteristic function of the interval [n, 0) for all n e N. Let f = 0 on R or f(r) = 0 for any r e R. Show that fn + f pointwise on R, that is, lim fn(r) = 0, Vr e R. %3D
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