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For each n ∈ N, define fn : [0, 1] → R byfn(x) =(xn cos(1/x) if x > 0,0 if x = 0.11. Prove that fn ∈ R[0, 1] for all n ∈ N

Question

For each n ∈ N, define fn : [0, 1] → R byfn(x) =(xn cos(1/x) if x > 0,0 if x = 0.11. Prove that fn ∈ R[0, 1] for all n ∈ N
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Solution 1

To prove that fn ∈ R[0, 1] for all n ∈ N, we need to show that fn is Riemann integrable on the interval [0, 1] for all natural numbers n.

Step 1: Define the function fn

For each natural number n, we define a function fn : [0, 1] → R as follows:

fn(x) = {xn cos(1/x) if x > 0, 0 if x = 0}

Step 2: Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.

Knowee AI  is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI  is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI  is a powerful AI-powered study tool designed to help you to solve study problem.
Knowee AI  is a powerful AI-powered study tool designed to help you to solve study problem.
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