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homework02

# homework02 - x n is bounded then lim U x n always exists 4...

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Math 5020 Homework 2 due Wednesday, February 23 1. Show that the following collection F is a filter over N : F = X N : n ̸∈ X 1 n + 1 < . 2. Let F be the filter of all cofinite sets over N . Show that for any infinite subset S N there is an ultrafilter G ⊇ F such that S ∈ G . 3. Let U be a non-principal ultrafilter over N . For any sequence ( x n ) of real numbers, define lim U x n = x if for any ϵ > 0, { n N : | x n - x | < ϵ } ∈ U. (a) Give an example of a sequence ( x n ) of real numbers so that lim U x n does not exist. (b) Show that if lim U x n exists then it is unique. (c) Show that if lim n x n exists then so does lim U x n and lim U x n = lim n x n . (d) Show that if (
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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 non-principle 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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