7.E: Valores Intermedios y Extremos (Ejercicios)
- Page ID
- 109524
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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.)
- \(S = \{\frac{1}{n} |n = 1,2,3,...\} \)
- \(T = \{r|r \text{ is rational and }r^2 < 2\}\)
- \((-∞,0) ∪ (1,∞)\)
- \(R = \{\frac{(-1)^n}{n} |n = 1,2,3,...\}\)
- \((2,3π] ∩ \mathbb{Q}\)
- El conjunto vacío\(\varnothing\)
Q3
Dejar\(S ⊆ R\) y dejar\(T = \{-x|x ∈ S\}\).
- Demostrar que\(b\) es un límite superior de\(S\) si y solo si\(-b\) es un límite inferior de\(T\).
- Demostrar que\(b = \sup S\) si y sólo si\(-b = \inf T\).