6 Questions
Give an alternate proof of the fact that every bounded sequence of complex numbers has a convergent subsequence.
By using the limes inferior.
Express the Reine-Borel property in an alternative manner.
Every collection of closed sets with an empty intersection contains a finite subcollection with empty intersection.
Prove using compactness that a closed bounded set of real numbers has a maximum.
Compactness guarantees the existence of a maximum in a closed bounded set of real numbers.
Provide an example showing that the intersection of a decreasing sequence of merely closed sets may be empty.
Show by example that the intersection of a decreasing sequence of merely closed sets can be empty.
Explain Cantor's lemma regarding the intersection of a decreasing sequence of nonempty compact sets.
The intersection of a decreasing sequence of nonempty compact sets is not empty.
Discuss the completeness of a space defined by sequences with a finite number of non-zero terms using a specific metric.
The space is complete, and the a-neighborhoods are not totally bounded.
Test your knowledge on finding the conjugate of harmonic functions through integration and explicit computation. Explore how to determine the conjugate function of specific harmonic functions like u = x^2 - y^2. Understand the process of deriving the most general conjugate function.
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