MHB Understanding the Cauchy-Schwarz Inequality

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The discussion focuses on providing examples to illustrate the Cauchy-Schwarz Inequality. Participants are encouraged to present a divergent sequence that is increasing but unbounded, as well as a divergent sequence that is bounded above but not increasing. These examples are essential to demonstrate that both conditions—being increasing and bounded—are necessary for the theorem's validity. The conversation highlights the importance of understanding these concepts in the context of the inequality. Clear examples will enhance comprehension of the theorem's application.
Joe20
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I am not sure what examples to give, need help on this. Have attached the theorem as well.
 

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You need to give an example of (i) a divergent sequence that is increasing but not bounded above, and (ii) a divergent sequence that is bounded above but not increasing. This will show that both conditions (increasing and bounded) are needed for the theorem to hold.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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