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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 (
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 ultralter. Let N be the ultraproduct U A n . (a) Give an explicit denition of U A n . In particular, dene 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 ....
View Full Document

Ask a homework question - tutors are online