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# 2.7.E: Problemas en los límites superior e inferior de las secuencias en$$E^{*}$$ (Exercises)

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## Ejercicio$$\PageIndex{1}$$

Completar los detalles faltantes en las pruebas de Teoremas 2 y$$3,$$ Corolario$$1,$$ y Ejemplos (a) y (b).

## Ejercicio$$\PageIndex{2}$$

Estado y probar los análogos de los Teoremas 1 y 2 y Corolario 2 para
$$\underline{\lim} x_{n}$$.

## Ejercicio$$\PageIndex{3}$$

Encontrar$$\overline{\lim } x_{n}$$ y$$\underline{\lim} x_{n}$$ si
(a)$$x_{n}=c$$ (constante);
(b)$$x_{n}=-n$$;
(c)$$x_{n}=n ;$$ y
(d)$$x_{n}=(-1)^{n} n-n$$
¿$$\lim x_{n}$$Existe en cada caso?

## Ejercicio$$\PageIndex{4}$$

$$\Rightarrow 4 .$$Se dice que una secuencia$$\left\{x_{n}\right\}$$ se agrupa en$$q \in E^{*},$$ y$$q$$ se llama su punto de clúster, iff cada uno$$G_{q}$$ contiene$$x_{n}$$ infinitamente muchos valores de$$n$$.
Mostrar que ambos$$\underline{L}$$ y$$\overline{L}$$ son los puntos de clúster$$(\underline{L} \text { the least and } \overline{L} \text { the }$$ más grandes).
[Pista: Utilice el teorema 2 y su análogo para$$\underline{L}$$.
Para demostrar que no$$p<\underline{L}$$ (o$$q>\overline{L} )$$ es un punto de cúmulo, asumir lo contrario y encontrar una contradicción al Corolario 2.]

## Ejercicio$$\PageIndex{5}$$

$$\Rightarrow 5 .$$Demostrar que
(i)$$\overline{\lim} \left(-x_{n}\right)=-\underline{\lim} x_{n}$$ y
(ii)$$\overline{\lim} \left(a x_{n}\right)=a \cdot \overline{\lim } x_{n}$$ si$$0 \leq a<+\infty$$.

## Ejercicio$$\PageIndex{6}$$

Demostrar que

\ [\ overline {\ lim} x_ {n} <+\ infty\ left (\ underline {\ lim} x_ {n} >-\ infty\ right)
\]
iff$$\left\{x_{n}\right\}$$ está delimitado arriba (abajo) en$$E^{1}$$.

## Ejercicio$$\PageIndex{7}$$

Demostrar que si$$\left\{x_{n}\right\}$$ y$$\left\{y_{n}\right\}$$ están delimitados en$$E^{1},$$ entonces

\ [\ overline {\ lim} x_ {n} +\ overline {\ lim} y_ {n}\ geq\ overline {\ lim}\ left (x_ {n} +y_ {n}\ right)\ geq\ overline {\ lim} x_ {n} +\ subrayado {\ lim} y_ {n}\ geq\ subrayado {\ lim} (x_ {n} + y_ {n})\ geq\ subrayado {\ lim} x_ {n } +\ subrayado {\ lim} y_ {n}.
\]
[Pista: Demostrar la primera desigualdad y luego usar eso y el Problema 5$$(\mathrm{i})$$ para los demás.]

## Ejercicio$$\PageIndex{8}$$

$$\Rightarrow 8 .$$Demostrar que si$$p=\lim x_{n}$$ en$$E^{1},$$ entonces

\ [\ subrayar {\ lim} (x_ {n} + y_ {n}) = p +\ subrayado {\ lim} y_ {n};
\] de
manera similar para$$\overline{L}$$.

## Ejercicio$$\PageIndex{9}$$

$$\Rightarrow 9 .$$Demostrar que si$$\left\{x_{n}\right\}$$ es monótona,$$\left\{x_{n}\right\} \uparrow,$$ entonces$$\lim x_{n}$$ existe$$i n E^{*} .$$ Específicamente, si entonces

\ [\ lim x_ {n} =\ sup _ {n} x_ {n},
\]
y si$$\left\{x_{n}\right\} \downarrow,$$ entonces

\ [\ lim x_ {n} =\ inf _ {n} x_ {n}.
\]

## Ejercicio$$\PageIndex{10}$$

$$\Rightarrow 10 .$$Demostrar que
(i) si lim$$x_{n}=+\infty$$ y$$(\forall n) x_{n} \leq y_{n},$$ luego también$$\lim y_{n}=+\infty,$$ y
(ii) si$$\lim x_{n}=-\infty$$ y$$(\forall n) y_{n} \leq x_{n},$$ entonces también$$\lim y_{n}=-\infty$$.

## Ejercicio$$\PageIndex{11}$$

Demostrar que si$$x_{n} \leq y_{n}$$ para todos$$n,$$ entonces

\ [\ subrayar {\ lim} x_ {n}\ leq\ subrayado {\ lim} y_ {n}\ text {y}\ overline {\ lim} x_ {n}\ leq\ overline {\ lim} y_ {n}.
\]

2.7.E: Problemas en los límites superior e inferior de las secuencias en$$E^{*}$$ (Exercises) is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.