Engineering Calculus Notes 185

Engineering Calculus Notes 185 - i,j > N...

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2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 173 equals sign (with zero on the other). Why is the given quantity non-negative? You may fnd it useFul to introduce some notation For di±erences oF coordinates, For example x 1 = x 2 x 1 x 2 = x 3 x 2 ; note that then x 1 + x 2 = x 3 x 1 . ) 5. Show that iF −→ p i L in R 3 , then { −→ p i } is bounded. 6. Suppose { −→ p i } is a sequence oF points in R 3 For which the distances between consecutive points Form a convergent series: s 0 dist( −→ p i , −→ p i +1 ) < . (a) Show that the sequence { −→ p i } is bounded. ( Hint: Use the triangle inequality) (b) Show that the sequence is Cauchy —that is, For every ε > 0 there exists N so that
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Unformatted text preview: i,j &gt; N guarantees dist( p i , p j ) &lt; . ( Hint: see ( Calculus Deconstructed , Exercise 2.5.9)) (c) Show that the sequence is convergent. 7. This problem concerns some properties oF accumulation points (Exercise 2 ). (a) Show that a sequence with at least two distinct accumulation points diverges. (b) Show that a bounded sequence has at least one accumulation point. (c) Give an example oF a sequence with no accumulation points. (d) Show that a bounded sequence with exactly one accumulation point converges to that point....
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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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