Unformatted text preview: x n ) is bounded then lim U x n always exists. 4. Consider the structure N = ( N , ,S ). Let A n = N for all n ∈ N . Let U be a nonprinciple ultraﬁlter. Let N ∗ be the ultraproduct ∏ U A n . (a) Give an explicit deﬁnition of ∏ U A n . In particular, deﬁne the zero element 0 N * and the successor function S N * . (b) Show that every nonzero element of N ∗ is an successor. (c) In N ∗ say an element a is standard if a = ( S N * ) n (0 N * ). Given an example of a nonstandard element of N ∗ ....
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 Spring '11
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 Set Theory, Sets, limU, limU xn, deﬁne limU xn

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