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P.4. Problem Set (Hard)

# P.4. Problem Set (Hard) - #1 In this problem we consider...

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#1 In this problem, we consider approximations of square roots. By the end, you will be able to construct a sequence that converges to any desired square root. Theorem (Monotone Convergence Theorem): A monotone sequence converges to a real number L if and only if it is bounded. Proof We need only consider the case where is decreasing. We already know that if a sequence converges to a real number L , then it is bounded. Hence, it is enough to prove that if is monotone and bounded, then it must be convergent. Exercise: Complete the proof of the theorem for the decreasing case. Let be the sequence given by and for all n . Exercise: Use induction to show that the sequence is bounded below. Once we establish that the sequence is bounded below, if it is monotone decreasing, we may conclude that the sequence converges to some limit. Calculating the first few terms, we find that , , , , This suggests the sequence is decreasing. To prove this, we must show , or equivalently, Exercise: Show that is equivalent to . Go on and prove for all n . We have now shown that the sequence is bounded and monotone decreasing. It follows from the monotone convergence theorem that the sequence converges to some limit L . We claim that . Exercise: Explain why . Use this and the uniqueness of limits to show that , i.e. . Exercise: Let k be some positive integer. Show that given by and converges to .

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#2 In this problem, we prove the Sequential Criterion for Functional Limits . We worked very hard to understand limits of sequences before we began discussing functional limits. In most calculus courses, the order is reversed. However, we will show there was indeed a reason for this. Theorem: Let f be a real valued function with domain D , and let p be an accumulation point of D . Then if and only if for every sequence of points in with , it follows that . (Note here that defines a sequence) Corollary: Let f be a real valued function with domain D and let p be a point in D . If p is an accumulation point, then f is continuous at p if and only if for every sequence of points in with , it follows that . In particular, this result can be useful in two ways, which we will illustrate after the proof is complete. We outline the proof, while you fill in the missing details. Proof
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P.4. Problem Set (Hard) - #1 In this problem we consider...

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