2.7.E: Problemas en los límites superior e inferior de las secuencias en\(E^{*}\) (Exercises)
- Page ID
- 114137
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Completar los detalles faltantes en las pruebas de Teoremas 2 y\(3,\) Corolario\(1,\) y Ejemplos (a) y (b).
Estado y probar los análogos de los Teoremas 1 y 2 y Corolario 2 para
\(\underline{\lim} x_{n}\).
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?
\(\Rightarrow 4 .\)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.]
\(\Rightarrow 5 .\)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\(0 \leq a<+\infty\).
Demostrar que
\ [\ overline {\ lim} x_ {n} <+\ infty\ left (\ underline {\ lim} x_ {n} >-\ infty\ right)
\]
iff\(\left\{x_{n}\right\}\) está delimitado arriba (abajo) en\(E^{1}\).
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.]
\(\Rightarrow 8 .\)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\(\overline{L}\).
\(\Rightarrow 9 .\)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}.
\]
\(\Rightarrow 10 .\)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\(\lim y_{n}=-\infty\).
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}.
\]