A sequence {ann21 contained in the interval [0, 1] is uniformly distributedeif and only iflim — [ f(a) = f f(x) dxnào nk=1-holds for any continuous function f(x) defined on [0, 1].

Answered: Image /qna-images/answer/b16b541d-2145-414d-bfe2-f2b418c7f4f7.jpg

Answered: A sequence {ann21 contained in the…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section6.4: Graphs Of Logarithmic Functions

Problem 60SE: Prove the conjecture made in the previous exercise.

See similar textbooks

Similar questions

Prove the conjecture made in the previous exercise.
For sequence of functions {nxe-nx} for x ∈ (0 + 1), what is the uniform norm of fn (x) - f(x) on (0 + x). is the sequence uniformly convergent?
Let fn(x) = x/(n^2+x^2) for x ∈ R. Show that the sequence {fn} converges uniformly to the function that is everywhere zero.
1. Let fgng be a sequence of integrable functions which converges a.e. to an integrable function g. Let ffng be a sequence of measurable functions such that jf j g and ff g converges
For any integer n ≥ 1 and any x ∈ (0,∞), define fn(x)= nx/(1+nx) (a) Let a > 0 be given. Prove that {fn} converges uniformly on the interval (a, ∞). (b) Prove that {fn} does not converge uniformly on (0,∞).
Suppose that a sequence of differentiable functions {fn} converges pointwiseto a function f on an interval [a,b], and the sequence {f′n}converges uniformlyto a function g on [a,b]. Then show that f is differentiable and f′(x) = g(x)on [a,b].
5. Suppose f is uniformly continuous on [n, n + 1] ∀ n ∈ N.Does it follow that f is uniformly continuous on R?
Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to ∞ in probability and Yn is bounded in probability. Show that Xn +Yn diverges to ∞ in probability.
Give an example of a sequence of differentiable functions {f_n: (-1,1)—>R} that converges uniformly but for which {f’_n(0)} is unbounded.
The series converges for every x in the half-open interval [−1, 1) but does not convergewhen x = 1. For a fixed x0 ∈ (−1, 1), explain how we can still use theWeierstrass M-Test to prove that f is continuous at x0.
Let (fn) be a sequence of differentiable functions defined on the closed interval [a, b], and assume (fn ) converges uniformly on [a, b]. If thereexists a point x0 ∈ [a, b] where fn(x0) is convergent, then (fn) converges uniformly on [a, b]. Proof. Exercise 6.3.7. Combining the last two results produces a stronger version of Theorem 6.3.1.
Let fn be a sequence of functions that are uniformly continuous on a set S. Suppose that (fn) converges uniformly to a function f on S. Prove that f is uniformly continuous on S.

Recommended textbooks for you

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

College Algebra

Algebra

ISBN:

9781938168383

Author:

Jay Abramson

Publisher:

OpenStax

SEE MORE TEXTBOOKS

A sequence {ann21 contained in the interval [0, 1] is uniformly distributede if and only if lim -- [ f(ak) = Σ I' f(x) dx n—c n k=1 -holds for any continuous function f(x) defined on [0, 1].