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

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Unformatted text preview: 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 iff U is closed under countable intersections. 2. Let (an )n∈N be a bounded real sequence. Prove that the following conditions are equivalent: (i) there exists a subsequence (ank )k∈N converging to b; (ii) there exists a free u.f. U on N with limU an = b. 3. Let U be a free u.f. on N and (an )n∈N be a bounded real sequence; prove that lim an ∈ [lim inf an , lim sup an ]. U n n 1 ...
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