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Chapter 2 Proofs

Chapter 2 Proofs - Pg 25 Proposition 2.1 The sequence...

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Pg 25, Proposition 2.1 – The sequence {1/n} converges to 0; that is, Pf: Let . Then by the definition of convergence there is a natural number N such that Therefore, when . By Corollary 1.10, we can say that there is a natural number N such that . So, Therefore, the sequence {1/n} converges to 0. QED Pg 25, Example 2.5 – Prove that the sequence does not converge. Pf: For purposes of contradiction let’s consider the sequence does converge to a real number a. We will let . By the definition of convergence there is a natural number N such that We can see that , so . But we also see that , so . Therefore, the sequence does not converge. QED Pg 26, Example 2.6 – Prove that the sequence converges to 3; that is, . Pf: Let . By the definition of convergence there is a natural number N such that Notice that . Notice that is a positive number. By Corollary 1.10, there is a natural number N such that or . Therefore, . So we have shown that sequence converges to 3. QED Pg 26, Proposition 2.2 – For any number c such that |c| < 1, the sequence converges to 0; that is, Pf: Let . By the definition of convergence there is a natural number N such that So, when . Let , then we can see that . Notice that d is a positive number, so by Bernoulli’s Inequality Notice that d ϵ is a positive number. So by Corollary 1.10, there is an natural number N such that or . So, Therefore, when |c| < 1, the sequence converges to 0. QED Pg 27, Lemma 2.3 – Every convergent sequence is bounded. Pf: Let be a sequence that converges to the number a. Taking , it follows from the definition of convergence that we may select a natural number N such that By using the Reverse Triangle Inequality, we have From which it follows that Define . Then, Therefore, every convergent sequence is bounded. QED Pg 27, Lemma 2.4 – Suppose that the sequence converges to the nonzero number b. Then there is a natural number N such that Pf: Notice that is a positive number. So, let . By the definition of convergence there is a natural number N such that

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By the Triangle Inequality we have With basic algebra can see that QED Pg 28, Theorem 2.5 – Suppose that the sequence converges to the number a and that the sequence converges to the number b. Then the sequence converges, and Also, the sequence converges, and Moreover, if for all n and , then the sequence converges, and Pf of (i): Let . By the definition of convergence we need a natural number N such that By the definition of convergence we know that there is a natural number N 1 such that We also know that there is a natural number N 2 such that Let . With the Triangle Inequality we see that Therefore (i) is true.
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