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Proving a Set E is Compact

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Please see the attached file for the fully formatted problems.

Let lambda n be a real decreasing sequence converging to

Prove E is compact if and only if = 0.

I am assuming compactness here refers to the sequential compactness. This
seems to make the most sense. Since this problem is an analysis problem,
please be sure to be rigorous, and include as much detail as possible so that
I can understand. Please also state if you are making use of some fact or
theorem. Thanks!

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Solution Summary

A set E is proven to be sequentially compact. The proof is detailed and well presented.

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