Mathematics · Real Analysis
Let &mathbb;N denote the set of all positive integers. For n ∈ &mathbb;N, let fn : [0, 1] → &mathbb;R be given by fn(x) = n(1 - nx) if 0 ≤ x ≤ 1/n, and 0 if x > 1/n. Which of the following statements is necessarily true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1The set {fn : n ∈ &mathbb;N} is compact in C[0, 1].
- 2The sequence of functions {fn}n ≥ 1 converges uniformly to the constant function 0.
- 3The sequence of functions {fn}n ≥ 1 does not converge pointwise on [0, 1].✓
- 4The sequence of functions {fn}n ≥ 1 is uniformly bounded.
Correct answer
The sequence of functions {fn}n ≥ 1 does not converge pointwise on [0, 1].
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