Question

Problem 2. Let $f_n: [a, b] \to \mathbb{R}$ be a sequence of integrable functions which uniformly converges to some $f: [a, b] \to \mathbb{R}$. Show that $f$ is integrable and give a counter example that the same property may fail if only pointwise convergence were assumed.

Name: Problem 2. Let fn: [a, b] →ℝ be a sequence of integrable functions which uniformly converges to some f: [a, b] →ℝ. Show that f is integrable and give a counter example that the same property may fail if only pointwise convergence were assumed.
Uploaded: 2023-01-05T03:47:38-08:00
Duration: 2 min 33 s
Channel: Madhur L
Description: Problem 2. Let fn: [a, b] →ℝ be a sequence of integrable functions which uniformly converges to some f: [a, b] →ℝ. Show that f is integrable and give a counter example that the same property may fail if only pointwise convergence were assumed.

          Problem 2. Let $f_n: [a, b] \to \mathbb{R}$ be a sequence of integrable functions which uniformly converges to some $f: [a, b] \to \mathbb{R}$. Show that $f$ is integrable and give a counter example that the same property may fail if only pointwise convergence were assumed.

Problem 2. Let fn: [a, b] →ℝ be a sequence of integrable functions which uniformly converges to some f: [a, b] →ℝ. Show that f is integrable and give a counter example that the same property may fail if only pointwise convergence were assumed.

Added by Emily C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

01/05/2023

Step 1

This means we need to show that the limit of the integrals of fn exists and is equal to the integral of f. Show more…

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Can you explain it clearly? Problem 2. Let fn : [a,b] → R be a sequence of integrable functions which uniformly converges to some f : [a,b] → R. Show that f is integrable and give a counter example that the same property may fail if only pointwise convergence were assumed.