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Miscellany4 - Analysis 2 Miscellaneous Problems 4 1 Let U...

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Analysis 2 Miscellaneous Problems 4 1. Let U be a free (non-principal) ultrafilter on Z ; define μ : P ( Z ) → { 0 , 1 } by μ ( A ) = 1 when A ∈ U and μ ( A ) = 0 when A < U . Prove that μ is σ -additive i ff U is closed under countable intersections. 2. Let ( a n ) n N be a bounded real sequence. Prove that the following conditions are equivalent: (i) there exists a subsequence (
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