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Dick said:
Oh so It doesn't make any difference at all. I see now.
It's just |a-b| = |b-a| looked wrong to me for a moment ( Can't even explain why). Now that that's clear I think I'm good.
Convergence of Sequences in [0,1]
- Thread starter STEMucator
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- Convergence Sequences
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SUMMARY
The discussion centers on the convergence of three sequences defined on the interval [0,1]: (i) \( s_n(x) = n^2x^2(1 - \cos(\frac{1}{nx})) \), (ii) \( s_n(x) = \frac{nx}{x+n} \), and (iii) \( s_n(x) = n\sin(\frac{x}{n}) \). The conclusions reached are that sequence (i) converges pointwise but not uniformly, while sequences (ii) and (iii) converge both pointwise and uniformly to their respective limits. The participants emphasize the importance of differentiability and bounded derivatives in establishing uniform convergence.
PREREQUISITES- Understanding of pointwise and uniform convergence in real analysis.
- Familiarity with limits and continuity of functions.
- Knowledge of differentiability and its implications for convergence.
- Proficiency in applying the epsilon-delta definition of limits.
- Study the epsilon-delta definition of uniform convergence in detail.
- Learn about the Weierstrass M-test for uniform convergence of sequences of functions.
- Explore the implications of differentiability on convergence properties.
- Investigate the relationship between compactness and uniform convergence in real analysis.
Mathematics students, particularly those studying real analysis, as well as educators and researchers interested in the properties of function sequences and convergence behavior.
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