Mathematics · Real Analysis
Let E = {1/n | n ∈ &DoubleStruckN;}. For each m ∈ &DoubleStruckN;, define fm : E → &Real; by: fm(x) = cos(mx) if x ≥ 1/m fm(x) = 0 if 1/(m+10) < x < 1/m fm(x) = x if x ≤ 1/(m+10) Then which of the following statements is true?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1There exists a subsequence of (fm)m≥1 which converges to 0 at every point of E
- 2There exist infinitely many subsequences of (fm)m≥1 which converge at every point of E✓
- 3Every subsequence of (fm)m≥1 converges at every point of E
- 4No subsequence of (fm)m≥1 converges at every point of E
Correct answer
There exist infinitely many subsequences of (fm)m≥1 which converge at every point of E
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