auxiliar8_MA1001_2012.pdf - INTRODUCCI\u00d3N AL C\u00c1LCULO Oto\u00f1o 2012 Profesor Ra\u00fal Uribe Auxiliar Braulio S\u00e1nchez Ib\u00e1\u00f1ez CLASE AUXILIAR 8 Axioma del

auxiliar8_MA1001_2012.pdf - INTRODUCCIÓN AL CÁLCULO...

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Unformatted text preview: INTRODUCCIÓN AL CÁLCULO Otoño 2012 Profesor: Raúl Uribe Auxiliar: Braulio Sánchez Ibáñez CLASE AUXILIAR 8 Axioma del Supremo P1. Demuestre las siguientes propiedades: (a) sup(A + a) = sup(A) + a (b) sup(cA) = c sup(A) (c) sup(A + B) = sup(A) + sup(B) P2. Demuestre las siguientes proposiciones: (a) Sean A, B conjuntos no vacíos de números reales, con A ⊆ B. Demuestre que ´ınf(B) ≤ ´ınf(A) ≤ sup(A) ≤ sup(B) (b) Sean S, T subconjuntos no vacíos de R tales que ∀x ∈ S, y ∈ T, x ≤ y. Pruebe que S tiene supremo, T tiene ínmo y que sup(S) ≤ ´ınf(T ). (c) Sean A, B,C subconjuntos no vacíos y acotados de R. Pruebe que si ∀x ∈ A, ∀y ∈ B, ∃z ∈ C tales que x + y ≤ z, entonces sup(A) + sup(B) ≤ sup(C)  P3.  1 Pruebe que ´ınf :n∈N =0 2n + 1 P4. Considere el conjunto A denido por   1 A= : n, m ∈ N, n 6= m |n − m| (a) Demuestre que m´ax(A) = 1 (b) Demuestre que ´ınf(A) = 0. HINT: Use la Propiedad Arquimediana. P5. Considere el conjunto A denido por: A = {x ≥ 0 : xn ≤ a} donde a ∈ (0, 1] y n > 2 es un natural jo. Pruebe que A posee supremo s, el cual satisface sn ≥ a. HINT: Use la siguiente propiedad: si b > 0 es tal que bn < a, entonces ∃c > b tal que bn < cn < a. ...
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