Engineering Calculus Notes 172

Engineering Calculus Notes 172 - coordinates. Lemma 2.3.5....

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160 CHAPTER 2. CURVES We will write −→ p i L in this case. An immediate corollary of the triangle inequality is the uniqueness of limits: Corollary 2.3.4. If a sequence { −→ p i } converges to L and also to L , then L = L . Proof. For any ε > 0, we can ±nd integers N and N so that dist( −→ p i ,L ) < ε for every i > N and also dist( −→ p i ,L ) < ε for every i > N . Pick an index i beyond both N and N , and estimate the distance from L to L as follows: dist( L,L ) dist( L, −→ p i ) + dist( −→ p i ,L ) < ε + ε = 2 ε. But this says that dist( L,L ) is less than any positive number and hence equals zero, so L = L by Equation ( 2.21 ). As a result of Corollary 2.3.4 , if −→ p i L we can refer to L as the limit of the sequence, and write L = lim −→ p i . A sequence is convergent if it has a limit, and divergent if it has none. The next result lets us relate convergence of points to convergence of their
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Unformatted text preview: coordinates. Lemma 2.3.5. Suppose { p i } is a sequence of points in R 3 with respective coordinates ( x i ,y i ,z i ) and L R 3 has coordinates ( 1 , 2 , 3 ) . Then the following are equivalent: 1. p i L (in R 3 ); 2. x i 1 , y i 2 , and z i 3 (in R ). Proof. (1) (2) : Suppose p i L . Given > 0, we can nd N so that i > N guarantees dist( p i ,L ) < . But then by Lemma 2.3.1 max( | x i 1 | , | y i 2 | , | z i 3 | ) < , showing that each of the coordinate sequences converges to the corresponding coordinate of L ....
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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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