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7.E: Valores Intermedios y Extremos (Ejercicios)

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    Q1

    Imita las definiciones de un límite superior de un conjunto y el límite inferior inferior (supremum) de un conjunto para dar definiciones para un límite inferior de un conjunto y el límite inferior mayor (infimum) de un conjunto.

    Nota: El infimum de un conjunto\(S\) se denota por\(\inf (S)\).

    Q2

    Encuentra el límite menor superior (supremum) y el mayor límite inferior (infimum) de los siguientes conjuntos de números reales, si existen. (Si uno no existe entonces díganlo.)

    1. \(S = \{\frac{1}{n} |n = 1,2,3,...\} \)
    2. \(T = \{r|r \text{ is rational and }r^2 < 2\}\)
    3. \((-∞,0) ∪ (1,∞)\)
    4. \(R = \{\frac{(-1)^n}{n} |n = 1,2,3,...\}\)
    5. \((2,3π] ∩ \mathbb{Q}\)
    6. El conjunto vacío\(\varnothing\)

    Q3

    Dejar\(S ⊆ R\) y dejar\(T = \{-x|x ∈ S\}\).

    1. Demostrar que\(b\) es un límite superior de\(S\) si y solo si\(-b\) es un límite inferior de\(T\).
    2. Demostrar que\(b = \sup S\) si y sólo si\(-b = \inf T\).

    This page titled 7.E: Valores Intermedios y Extremos (Ejercicios) was last modified on Sun, 30 Oct 2022 13:47:41 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Eugene Boman and Robert Rogers (OpenSUNY) via source content that was edited to the style and standards of the LibreTexts platform.