**4.** Let \( f_n(x) = \frac{nx}{1+n^2x^2}, x \in \mathbb{R}. \) Prove that \( f_n \to 0 \) pointwise on \( \mathbb{R} \), then check whether \( f_n \to 0 \) uniformly or not on \([0,1]\).**5.** Same \( f_n \) in Q4, check whether \( f_n \to 0 \) uniformly or not on \([1,\infty)\).

Since you have posted a multiple question according to guidelines I will solve first(Q4) question…

Answered: nx 4. Let fn(x) = 1+7²², x € R. Prove…

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nx 4. Let fn(x) = 1+7²², x € R. Prove that fn → 0 pointwise on R, then check whether fn → 0 uniformly or not on [0, 1]

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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**4.** Let \( f_n(x) = \frac{nx}{1+n^2x^2}, x \in \mathbb{R}. \) Prove that \( f_n \to 0 \) pointwise on \( \mathbb{R} \), then check whether \( f_n \to 0 \) uniformly or not on \([0,1]\).

**5.** Same \( f_n \) in Q4, check whether \( f_n \to 0 \) uniformly or not on \([1,\infty)\).

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