analisis 5 - Sucesiones de nmeros reales u 26 de marzo de...

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Sucesiones de n´umeros reales 26 de marzo de 2011 1. Generales 1. Demuestre que si l´ ım n →∞ x n = a , entonces l´ ım n →∞ | x n | = | a | .¿Es verdadero el rec´ ıproco? Demostraci´on. Sabemos que: ım n →∞ x n = a (1) De (1) sabemos que sea ε 1 > 0 existe n 1 R para todo n > n 0 tal que: | x n - a | < ε 1 (2) Debo probar que sea ε > 0 existe n 0 R para todo n > n 0 tal que: || x n | - | a || < ε (3) Como: | x n | = | x n - a + a | | x n - a + a | ≤ | x n - a | + | a | | x n | - | a | ≤ | x n - a | (4) Analogamente : | a | = | a - x n + x n | | a | ≤ | a - x n | + | x n | | a | - | x n | ≤ | a - x n | (5) De (4) y (5), por la definici´on de valor absoluto tenemos que: || x n | - | a || ≤ max {| x n | - | a | , | a | - | x n |} Entonces tenemos que: || x n | - | a || ≤ | x n - a | (6) De aqui tenemos que : || x n | - | a || ≤ ε 1 Por lo que basta tomar n 0 = n 1 1
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El reciproco no es verdadero ya que si tomamos la suceci´on ( - 1) n + 1, el l´ ım n
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This note was uploaded on 03/27/2011 for the course MATHEMATIC 504 taught by Professor Carlostrujillo during the Winter '09 term at Buena Vista.

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analisis 5 - Sucesiones de nmeros reales u 26 de marzo de...

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