Engineering Calculus Notes 174

Engineering Calculus Notes 174 - x i k 1 R . Now, consider...

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162 CHAPTER 2. CURVES Axiom 2.3.2) directly to R 3 . However, the Bolzano-Weierstrass Theorem ( Calculus Deconstructed , Prop. 2.3.8), which is an efective substitute For the Completeness Axiom, can easily be extended From sequences oF numbers to sequences oF points: Proposition 2.3.7 (Bolzano-Weierstrass Theorem) . Every bounded sequence of points in R 3 has a convergent subsequence. Proof. Suppose M is an upper bound on distances From the origin: dist( −→ p i , O ) < M For all i. In particular, by Lemma 2.3.1 , | x i | < M For all i so the ±rst coordinates oF our points Form a bounded sequence oF numbers. But then the Bolzano-Weierstrass Theorem in R says that we can pick a convergent subsequence oF these numbers, say
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Unformatted text preview: x i k 1 R . Now, consider the (sub)sequence oF points { p i k } , and look at their second coordinates. We have | y i k | < M For all i k and hence passing to a sub-(sub)sequence, we have y i k 2 R . Note that passing to a subsequence does not afect convergence oF the Frst coordinates: x i k 1 . In a similar way, the third coordinates are bounded v v v z i k v v v < M For all i k and hence we can nd a convergent sub-(sub-sub)sequence { p i k } For which the third coordinates converge as well: z i k 3 R ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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