7. Consider the set Seq[ℕ] of all the infinite sequences (a_k)_{k ∈ ℕ} = (a_0, a_1, a_2, a_3, ...) of natural numbers. A sequence (a_k)_{k ∈ ℕ} is constant if there is some number n ∈ ℕ so that a_k = n, for all k ∈ ℕ. (a) Construct an injection i: ℕ → Seq[ℕ]. (b) Prove that there is no surjection f: ℕ → Seq[ℕ]. (Consider the arguments of the proof of Cantor’s Theorem that we saw in class.) (c) Deduce from above that ℵ₀ < |Seq[ℕ]|. (Actually, it can be proven that |Seq[ℕ]| = |ℝ|.)

7. Consider the set Seq[ℕ] of all the infinite sequences (a_k)_{k ∈ ℕ} = (a_0, a_1, a_2, a_3, ...) of natural numbers. A sequence (a_k)_{k ∈ ℕ} is constant if there is some number n ∈ ℕ so that a_k = n, for all k ∈ ℕ. (a) Construct an injection i: ℕ → Seq[ℕ]. (b) Prove that there is no surjection f: ℕ → Seq[ℕ]. (Consider the arguments of the proof of Cantor’s Theorem that we saw in class.) (c) Deduce from above that ℵ₀ < |Seq[ℕ]|. (Actually, it can be proven that |Seq[ℕ]| = |ℝ|.)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 41E

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