4130hw4

# 4130hw4 - (Hint see p 75 of the textbook(5 Find if they...

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4130 HOMEWORK 4 Due Tuesday March 2 (1) Let N N denote the set of all sequences of natural numbers. That is, N N = { ( a 1 ,a 2 ,a 3 ,... ) : a i N } . Show that | N N | = |P ( N ) | . (2) Let { x n } be a Cauchy sequence of rational numbers. Regarding { x n } as a sequence of real numbers, show that { x n } converges to the real number x deﬁned as the equivalence class of the sequence { x n } . (3) Section 2.2.4 # 4. (4) Show that every subset S of R which is bounded below has a greatest lower bound.
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Unformatted text preview: (Hint: see p. 75 of the textbook.) (5) Find, if they exist, the supremum (least upper bound) and inﬁmum (greatest lower bound) of the following subsets of R . (a) { 1 , 2 , 3 } . (b) (0 , 1) ∪ { 2 } ∪ [3 , 4) = { x ∈ R : 0 < x < 1 or x = 2 or 3 ≤ x < 4 } . (c) { 1-1 n : n ∈ N } . (d) Q . (6) Prove Theorem 2.3.2 in the textbook. 1...
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