Sucesiones 3 - (xn es la sucesi´n definida como o(xn =...

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Unformatted text preview: (xn ) es la sucesi´n definida como: o (xn ) = 1 + (−1)n+1 P.D. Que xn no es convergente. Supongamos que xn es convergente, tenemos que l´ xn = a entonces toda subsucesi´n de (xn ) ım o converge a a. Sea (xnk ) = (0, 0, 0, ...) y (xn ) = (2, 2, 2, ...) subsucesiones de (xn ): k (xnk ) = (0, 0, 0, ...): x:N→R n → xn = 1 + (−1)n+1 α:N→N k → 2k x◦α:N→R k →= x ◦ α = x(α(k )) = x(2k ) = 1 + (−1)2k+1 = 1 + (−1) = 0 (xnk ) = (2, 2, 2, ...): x:N→R n → xn = 1 + (−1)n+1 β:N→N k → 2k − 1 x◦β :N→R k →= x ◦ β = x(β (k )) = x(2k ) = 1 + (−1)2k−1+1 = 1 + (−1)2k = 2 Los L´ ımites de cada subsucesi´n son: o l´ xnk = 0 ım l´ xn ım k =2 Por: L´ ımite de una constante es la misma constante: P.D.l´ xn = a sea a una constante: ım Sea ε > 0, debemos encontrar un n0 ∈ N tal que: |xn − a| < ε para n ≥ n0 : |xn − a| = |a − a| = 0 < ε Sea n0 = 1 P.D. |xn − a| < ε 1 Para todo n ≥ 1 |xn − a| = |a − a| = |0| = 0 < ε Tenemos que: l´ xnk = 0 ım l´ xn ım k =2 como las dos subsucesiones convergen a distintos l´ ımtes podemos decir que llegamos a un absurdo por lo que la sucesi´n no converge. o 2 ...
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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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Sucesiones 3 - (xn es la sucesi´n definida como o(xn =...

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