HW4Hint - | a n i-a | ≥ ± for all i 2 Explain why a n i...

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Hint for Exercise 2.5.4 This problem requires an argument in several stages. Here is an outline. You need to write it out more completely and supply the explanations needed for each step. 1. The proof will be by contradiction. Suppose ( a n ) does not converge to a . By negating the definition of convergence, explain why this means that for some > 0 there is a subsequence with
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Unformatted text preview: | a n i-a | ≥ ± for all i . 2. Explain why ( a n i ) must have a convergent subsequence: ( a n i j ) → ‘ . 3. Explain why ‘ 6 = a . 4. Explain why a subsequence of a subsequence is still a subsequence of the original ( a n ), and so step 3 above contradicts the hypotheses of the problem. 5. This completes the proof by contradiction....
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