This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Analysis 2
Miscellaneous Problems 4
1. Let U be a free (nonprincipal) 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 σadditive 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 ...
View Full
Document
 Spring '09
 LARSON

Click to edit the document details