Commit inicial, primer cuatrimestre.

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+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures false
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Definición axiomática de 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ es el cuerpo conmutativo totalmente ordenado y completo.
+\end_layout
+
+\begin_layout Subsection
+Cuerpo conmutativo
+\end_layout
+
+\begin_layout Standard
+Conjunto con dos operaciones internas: suma (
+\begin_inset Formula $\mathbb{K}\times\mathbb{K}\rightarrow\mathbb{K}$
+\end_inset
+
+ con 
+\begin_inset Formula $(x,y)\mapsto x+y$
+\end_inset
+
+) y producto (
+\begin_inset Formula $\mathbb{K}\times\mathbb{K}\rightarrow\mathbb{K}$
+\end_inset
+
+ con 
+\begin_inset Formula $(x,y)\mapsto x\cdot y$
+\end_inset
+
+), con las siguientes propiedades: 
+\begin_inset Formula $\forall a,b,c\in\mathbb{K}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Asociativa de la suma: 
+\series default
+
+\begin_inset Formula $a+(b+c)=(a+b)+c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Conmutativa de la suma: 
+\begin_inset Formula $a+b=b+a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Elemento neutro para la suma
+\series default
+ o 
+\series bold
+nulo:
+\series default
+ 
+\begin_inset Formula $\exists!0\in\mathbb{K}:\forall a\in\mathbb{K},0+a=a$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Pongamos que existe otro 
+\begin_inset Formula $0$
+\end_inset
+
+ (
+\begin_inset Formula $0'$
+\end_inset
+
+), entonces 
+\begin_inset Formula $0=0+0'=0'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Inverso para la suma
+\series default
+ u 
+\series bold
+opuesto:
+\series default
+ 
+\begin_inset Formula $\exists!a':a+a'=0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $a':=-a$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Pongamos que existe otro opuesto 
+\begin_inset Formula $a''$
+\end_inset
+
+, entonces 
+\begin_inset Formula $a'=0+a'=(a''+a)+a'=a''+(a+a')=a''+0=a''$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Asociativa del producto:
+\series default
+ 
+\begin_inset Formula $a\cdot(b\cdot c)=(a\cdot b)\cdot c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Conmutativa del producto:
+\series default
+ 
+\begin_inset Formula $a\cdot b=b\cdot a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Elemento neutro para el producto
+\series default
+ o 
+\series bold
+unidad:
+\series default
+ 
+\begin_inset Formula $\exists!1\in\mathbb{K}:\forall a\in K,1\cdot a=a$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Pongamos que existe otro 
+\begin_inset Formula $1$
+\end_inset
+
+ (
+\begin_inset Formula $1'$
+\end_inset
+
+), entonces 
+\begin_inset Formula $1=1\cdot1'=1'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Inverso para el producto:
+\series default
+ 
+\begin_inset Formula $\forall a\in\mathbb{K}\backslash\{0\},\exists!a'':a\cdot a''=1$
+\end_inset
+
+; 
+\begin_inset Formula $a'':=\frac{1}{a}:=a^{-1}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Pongamos que existe otro 
+\begin_inset Formula $a''$
+\end_inset
+
+ (
+\begin_inset Formula $a'$
+\end_inset
+
+), entonces 
+\begin_inset Formula $a''=1\cdot a''=(a'\cdot a)\cdot a''=a'\cdot(a\cdot a'')=a'\cdot1=a'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Distributiva:
+\series default
+ 
+\begin_inset Formula $a\cdot(b+c)=a\cdot b+a\cdot c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+De aquí podemos deducir que:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a=b\iff a-b=0$
+\end_inset
+
+; 
+\begin_inset Formula $b\neq0\implies(a=b\iff a\cdot b^{-1}=1)$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+a=b\iff a+(-b)=b+(-b)\iff a-b=0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+b\neq0\implies\exists b^{-1}\implies(a=b\iff a\cdot b^{-1}=b\cdot b^{-1}=1)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\cdot0=0$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+a\cdot0+0=a\cdot0=a\cdot(0+0)=a\cdot0+a\cdot0\implies-a\cdot0+a\cdot0=-a\cdot0+a\cdot0+a\cdot0\implies0=a\cdot0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(-a)\cdot b=-(ab)$
+\end_inset
+
+; 
+\begin_inset Formula $(-1)\cdot a=-a$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+(-a)\cdot b+a\cdot b=(-a+a)\cdot b=0\cdot b=0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+(-1)\cdot a=-(1\cdot a)=-a
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Totalmente ordenado
+\end_layout
+
+\begin_layout Standard
+Aquel con relación binaria 
+\begin_inset Formula $\leq$
+\end_inset
+
+ con las siguientes propiedades: 
+\begin_inset Formula $\forall x,y,z\in\mathbb{K}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Reflexiva:
+\series default
+ 
+\begin_inset Formula $x\leq x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Antisimétrica:
+\series default
+ 
+\begin_inset Formula $x\leq y\land y\leq x\iff x=y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Transitiva:
+\series default
+ 
+\begin_inset Formula $x\leq y\land y\leq z\implies x\leq z$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Orden total:
+\series default
+ 
+\begin_inset Formula $x\leq y\lor y\leq x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $x\leq y\implies x+z\leq y+z$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $x\leq y\land0\leq z\implies x\cdot z\leq y\cdot z$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una relación binaria que cumple las propiedades 1–3 se denomina de 
+\series bold
+orden.
+
+\series default
+ Si también cumple (4), de 
+\series bold
+orden total.
+
+\series default
+ El conjunto de todas definen un 
+\series bold
+cuerpo totalmente ordenado.
+\end_layout
+
+\begin_layout Standard
+Notación: 
+\begin_inset Formula $x<y\iff y>x\iff x\leq y\land x\neq y$
+\end_inset
+
+; 
+\begin_inset Formula $x\geq y\iff y\leq x$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Podemos deducir que:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $c<0\iff-c>0$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+c<0\iff c+(-c)<-c\iff0<-c
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\leq b\land c\leq d\implies a+c\leq b+d$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+\begin{array}{c}
+a\leq b\implies a+c\leq b+c\\
+c\leq d\implies b+c\leq b+d
+\end{array}\implies a+c\leq b+d
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\leq b\iff-a\geq-b$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+a\leq b\iff a+(-a)+(-b)\leq b+(-b)+(-a)\iff-b\leq-a
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $c<0\implies(a\leq b\iff ca\geq cb)$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+c<0\implies-c>0\implies(-c)a\leq(-c)b\implies-(ca)\leq-(cb)\implies ca\geq cb
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a\neq0\implies a\cdot a>0$
+\end_inset
+
+; 
+\begin_inset Formula $1\neq0\implies1\geq0$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+a\cdot a\neq0;\ \begin{cases}
+a\geq0 & \implies a\cdot a\geq a\cdot0=0\\
+a\leq0 & \implies a\cdot a\geq a\cdot0=0
+\end{cases}\implies a\cdot a>0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+0\neq1\land1=1\cdot1\implies1>0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $a>0\iff a^{-1}>0$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Supongamos 
+\begin_inset Formula $a^{-1}\leq0$
+\end_inset
+
+ y 
+\begin_inset Formula $a>0$
+\end_inset
+
+.
+ Entonces, 
+\begin_inset Formula $1=a\cdot a^{-1}\leq0$
+\end_inset
+
+.
+ Pero 
+\begin_inset Formula $1\nleq0\#$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $b>0\implies(a\leq b\implies a^{-1}\leq b^{-1})$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+a\geq b\implies a^{-1}\cdot a\geq b\cdot a^{-1}\implies1\geq b\cdot a^{-1}\implies b^{-1}\geq b^{-1}(b\cdot a^{-1})=a^{-1}\implies b^{-1}\geq a^{-1}
+\]
+
+\end_inset
+
+El recíproco es cierto si 
+\begin_inset Formula $a>0$
+\end_inset
+
+ también, pues en el último paso multiplicaríamos por 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Completo
+\end_layout
+
+\begin_layout Standard
+Aquel que cumple el 
+\series bold
+axioma del supremo:
+\series default
+ todo subconjunto no vacío de 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ acotado superiormente tiene supremo.
+ Un conjunto 
+\begin_inset Formula $\emptyset\neq A\subseteq\mathbb{R}$
+\end_inset
+
+ está acotado superiormente si 
+\begin_inset Formula $\exists M\in\mathbb{R}:\forall a\in A,a\leq M$
+\end_inset
+
+, entonces 
+\begin_inset Formula $M$
+\end_inset
+
+ es cota superior de 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\alpha\in\mathbb{R}$
+\end_inset
+
+ es el supremo de 
+\begin_inset Formula $A$
+\end_inset
+
+ (
+\begin_inset Formula $\alpha=\sup A$
+\end_inset
+
+) si es su menor cota superior, y cumple que 
+\begin_inset Formula $\forall\varepsilon>0,\exists a\in A:\alpha-\varepsilon<a\leq\alpha$
+\end_inset
+
+.
+ Cuando 
+\begin_inset Formula $\alpha\in A$
+\end_inset
+
+, se le llama también máximo.
+\end_layout
+
+\begin_layout Standard
+Igualmente, un subconjunto 
+\begin_inset Formula $\emptyset\neq A\subseteq\mathbb{R}$
+\end_inset
+
+ está acotado inferiormente si 
+\begin_inset Formula $\exists M\in\mathbb{R}:\forall a\in A,M\leq a$
+\end_inset
+
+, entonces 
+\begin_inset Formula $M$
+\end_inset
+
+ es cota inferior de 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\alpha\in\mathbb{R}$
+\end_inset
+
+ es el ínfimo de 
+\begin_inset Formula $A$
+\end_inset
+
+ (
+\begin_inset Formula $\alpha=\inf A$
+\end_inset
+
+) si es su mayor cota inferior.
+ Todo cuerpo que verifica el axioma del supremo también cumple que todo
+ subconjunto no vacío acotado inferiormente tiene ínfimo.
+ 
+\series bold
+Demostración:
+\series default
+ si 
+\begin_inset Formula $A$
+\end_inset
+
+ está acotado inferiormente por 
+\begin_inset Formula $\alpha$
+\end_inset
+
+, 
+\begin_inset Formula $-A=\{-a\}_{a\in A}$
+\end_inset
+
+ está acotado superiormente por 
+\begin_inset Formula $-\alpha$
+\end_inset
+
+, y si 
+\begin_inset Formula $\beta$
+\end_inset
+
+ es su supremo, entonces 
+\begin_inset Formula $-\beta$
+\end_inset
+
+ será el ínfimo de 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Otras propiedades de los números (
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+, 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+Un subconjunto 
+\begin_inset Formula $I\subseteq\mathbb{K}$
+\end_inset
+
+ es
+\series bold
+ inductivo
+\series default
+ si 
+\begin_inset Formula $1\in I$
+\end_inset
+
+ y 
+\begin_inset Formula $n\in I\implies n+1\in I$
+\end_inset
+
+.
+ Todo cuerpo o intersección de conjuntos inductivos es un conjunto inductivo.
+ Ahora tomemos el 
+\begin_inset Quotes cld
+\end_inset
+
+bicho
+\begin_inset Quotes crd
+\end_inset
+
+ 
+\begin_inset Formula $\bigcap\{I:I\text{ es un conjunto inductivo de }\mathbb{R}\}$
+\end_inset
+
+, la intersección de todos los conjuntos inductivos y por tanto el más pequeño
+ de ellos.
+ Así, el conjunto de 
+\series bold
+números naturales
+\series default
+ 
+\begin_inset Formula $\mathbb{N}:=\text{bicho}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Podemos definir 
+\begin_inset Formula $2=1+1$
+\end_inset
+
+, 
+\begin_inset Formula $3=2+1$
+\end_inset
+
+, 
+\begin_inset Formula $4=3+1$
+\end_inset
+
+, etc.
+ Propiedades 
+\begin_inset Quotes cld
+\end_inset
+
+obvias
+\begin_inset Quotes crd
+\end_inset
+
+ de los naturales:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall n<1,n\notin\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall n\in\mathbb{N},\nexists x\in\mathbb{N}:n<x<n+1$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Para 
+\begin_inset Formula $n=1$
+\end_inset
+
+: Suponemos 
+\begin_inset Formula $\exists r\in\mathbb{N}:1<r<2=1+1$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $S=\{n\in\mathbb{N}:1<n<2\}\neq\emptyset\land r\in s$
+\end_inset
+
+.
+ Sabemos que 
+\begin_inset Formula $1\in\mathbb{N}\backslash S$
+\end_inset
+
+.
+ Consideremos un número 
+\begin_inset Formula $m\in\mathbb{N}\backslash S$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $m\leq1\lor m\geq2$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\begin{array}{cc}
+n\leq1\implies & n=1\implies n+1=2\in\mathbb{N}\backslash S\\
+n\geq2\implies & n+1\geq2+1=3,\,n+1\in\mathbb{N}\backslash S
+\end{array}\implies\mathbb{N}\backslash S=\mathbb{N}\implies S=\emptyset
+\]
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Demostrar resto de propiedades cuando las estudiemos, si no como ejercicio.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall n,m\in\mathbb{N},n+m\in\mathbb{N}\land n\cdot m\in\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall n,m\in\mathbb{N},m>n\implies\exists k\in\mathbb{N}:m=n+k$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Definimos 
+\begin_inset Formula $\mathbb{Z}:=\{0\}\cup\{n\in\mathbb{R}:n\in\mathbb{N}\text{ o }-n\in\mathbb{N}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{Q}:=\{m\cdot n^{-1}:m\in\mathbb{Z},n\in\mathbb{N}\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Método de inducción
+\end_layout
+
+\begin_layout Standard
+Método de demostración basado en definir un conjunto 
+\begin_inset Formula $S\subseteq\mathbb{N}$
+\end_inset
+
+ que cumpla la propiedad 
+\begin_inset Formula $P(n)$
+\end_inset
+
+ a demostrar en 
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+ y demostrar que es inductivo.
+ Como 
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+ es el conjunto inductivo más pequeño, tenemos 
+\begin_inset Formula $S=\mathbb{N}$
+\end_inset
+
+.
+ Para demostrar esto:
+\end_layout
+
+\begin_layout Enumerate
+Comprobamos que 
+\begin_inset Formula $P(1)$
+\end_inset
+
+ es verdad.
+\end_layout
+
+\begin_layout Enumerate
+Demostramos que 
+\begin_inset Formula $P(n)\implies P(n+1)$
+\end_inset
+
+.
+ Para ello, demostramos 
+\begin_inset Formula $P(n+1)$
+\end_inset
+
+ tomando como propiedad 
+\begin_inset Formula $P(n)$
+\end_inset
+
+ (la 
+\series bold
+hipótesis de inducción
+\series default
+).
+\end_layout
+
+\begin_layout Standard
+Dado un número natural 
+\begin_inset Formula $N$
+\end_inset
+
+, un conjunto 
+\begin_inset Formula $S\subseteq\{n\in\mathbb{N}:n\geq N\}\subseteq\mathbb{N}$
+\end_inset
+
+ nos sirve para realizar demostraciones para los naturales a partir de un
+ número arbitrario.
+ Por último, la 
+\series bold
+versión fuerte
+\series default
+ del método de inducción nos permite definir 
+\begin_inset Formula $S$
+\end_inset
+
+ tal que 
+\begin_inset Formula $1\in S$
+\end_inset
+
+ y 
+\begin_inset Formula $1,2,\dots,n\in S\implies n+1\in S$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $S=\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+De esta forma podemos demostrar el 
+\series bold
+Teorema Fundamental de la Aritmética
+\series default
+, que nos dice que todo número entero 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+ es primo o producto de primos.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $A=\{2\leq n\in\mathbb{N}:n\text{ cumple el Teorema Fund. de la Aritmética}\}$
+\end_inset
+
+.
+ Sabemos que 
+\begin_inset Formula $2\in A$
+\end_inset
+
+, y queremos demostrar que, si tenemos un 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $2,3,\dots,n\in A$
+\end_inset
+
+, entonces 
+\begin_inset Formula $n+1\in A$
+\end_inset
+
+.
+ Ahora, o bien 
+\begin_inset Formula $n+1$
+\end_inset
+
+ es primo, en cuyo caso 
+\begin_inset Formula $n+1\in A$
+\end_inset
+
+, o no lo es, pero entonces 
+\begin_inset Formula $\exists p,q\in\mathbb{N}:1<p,q<n+1:p\cdot q=n+1$
+\end_inset
+
+, y como hemos supuesto que 
+\begin_inset Formula $2,3,\dots,n\in A$
+\end_inset
+
+, entonces 
+\begin_inset Formula $p,q\in A$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $p$
+\end_inset
+
+ y 
+\begin_inset Formula $q$
+\end_inset
+
+ son primos o producto de primos, 
+\begin_inset Formula $n+1$
+\end_inset
+
+ también lo es, por lo que 
+\begin_inset Formula $n+1\in A$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Propiedad arquimediana
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ cumple la 
+\series bold
+propiedad arquimediana:
+\series default
+ 
+\begin_inset Formula $\forall0<y,x\in\mathbb{R},\exists n\in\mathbb{N}:x<ny$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ De no ser así, 
+\begin_inset Formula $A:=\{ny:n\in\mathbb{N}\}$
+\end_inset
+
+ estaría acotado superiormente por 
+\begin_inset Formula $x$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $\alpha:=\sup A$
+\end_inset
+
+; tendríamos que 
+\begin_inset Formula $\forall n\in\mathbb{N},ny\leq\alpha$
+\end_inset
+
+.
+ Por otro lado, 
+\begin_inset Formula $\alpha-y$
+\end_inset
+
+ no sería cota superior de 
+\begin_inset Formula $A$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\exists n_{0}\in\mathbb{N}:\alpha-y<n_{0}y$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $\alpha<(n_{0}+1)y$
+\end_inset
+
+, lo que contradice el hecho de que 
+\begin_inset Formula $A$
+\end_inset
+
+ esté acotado superiormente por 
+\begin_inset Formula $\alpha$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Standard
+Por tanto 
+\begin_inset Formula $\mathbb{N}$
+\end_inset
+
+ no está acotado superiormente, y 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ no está acotado superior ni inferiormente.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Principio de la buena ordenación:
+\series default
+ Todo subconjunto no vacío 
+\begin_inset Formula $A\subseteq\mathbb{N}$
+\end_inset
+
+ tiene 
+\series bold
+primer elemento
+\series default
+.
+ 
+\series bold
+Demostración:
+\series default
+ supongamos que 
+\begin_inset Formula $A$
+\end_inset
+
+ no tuviera primer elemento y sea 
+\begin_inset Formula $B:=\mathbb{N}\backslash A$
+\end_inset
+
+ el complementario de 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $1\notin A$
+\end_inset
+
+, pues de lo contrario tendría primer elemento; por tanto 
+\begin_inset Formula $1\in B$
+\end_inset
+
+.
+ Además, si 
+\begin_inset Formula $1,\dots,n\in B$
+\end_inset
+
+ entonces 
+\begin_inset Formula $n+1\in B$
+\end_inset
+
+, pues de lo contrario tendríamos que 
+\begin_inset Formula $n+1\in A$
+\end_inset
+
+ sería el primer elemento.
+ Por tanto 
+\begin_inset Formula $B=\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $A=\emptyset$
+\end_inset
+
+.
+\begin_inset Formula $\#$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $x\in\mathbb{R}$
+\end_inset
+
+, llamamos 
+\series bold
+parte entera
+\series default
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+ o 
+\begin_inset Formula $[x]$
+\end_inset
+
+ al único 
+\begin_inset Formula $m\in\mathbb{Z}$
+\end_inset
+
+ que verifica 
+\begin_inset Formula $m\leq x<m+1$
+\end_inset
+
+.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Demostremos que existe.
+ Supongamos que 
+\begin_inset Formula $x\geq1$
+\end_inset
+
+.
+ Aplicando la propiedad arquimediana a 
+\begin_inset Formula $x$
+\end_inset
+
+ y 
+\begin_inset Formula $1$
+\end_inset
+
+, se tiene que el conjunto 
+\begin_inset Formula $\{n\in\mathbb{N}:n>x\}\neq\emptyset$
+\end_inset
+
+, por lo que tiene un primer elemento 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+.
+ Si tomamos 
+\begin_inset Formula $m:=k-1$
+\end_inset
+
+ obtenemos el resultado.
+ La unicidad se debe a que no existe ningún número natural entre 
+\begin_inset Formula $1$
+\end_inset
+
+ y 
+\begin_inset Formula $2$
+\end_inset
+
+ y, por inducción, tampoco entre 
+\begin_inset Formula $m$
+\end_inset
+
+ y 
+\begin_inset Formula $m+1$
+\end_inset
+
+ para ningún 
+\begin_inset Formula $m\in\mathbb{N}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+De aquí podemos obtener que 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ es 
+\series bold
+denso
+\series default
+ en 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+, es decir, que si 
+\begin_inset Formula $x,y\in\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $x<y$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\exists r\in\mathbb{Q}:x<r<y$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Por la propiedad arquimediana 
+\begin_inset Formula $\exists n\in\mathbb{N}:1<n(y-x)$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\frac{1}{n}<y-x$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $m:=[nx]$
+\end_inset
+
+, entonces 
+\begin_inset Formula $m\leq nx<m+1$
+\end_inset
+
+, por lo que
+\begin_inset Formula 
+\[
+\frac{m}{n}\leq x<\frac{m+1}{n}=\frac{m}{n}+\frac{1}{n}\leq x+\frac{1}{n}<x+(y-x)=y
+\]
+
+\end_inset
+
+Tomamos 
+\begin_inset Formula $r=\frac{m+1}{n}$
+\end_inset
+
+ para obtener el resultado buscado.
+\end_layout
+
+\begin_layout Subsection
+Raíces cuadradas
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $x=y^{2}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $y$
+\end_inset
+
+ es una 
+\series bold
+raíz cuadrada
+\series default
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $-y$
+\end_inset
+
+ también lo es, y 
+\begin_inset Formula $x$
+\end_inset
+
+ no puede tener más raíces cuadradas.
+ Definimos
+\begin_inset Formula 
+\[
+\sqrt{x}:=\sup\{0\leq r\in\mathbb{Q}:r^{2}<x\}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+No existe ningún número racional cuyo cuadrado sea 
+\begin_inset Formula $2$
+\end_inset
+
+.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Supongamos 
+\begin_inset Formula $\exists p,q\in\mathbb{N}:\frac{p^{2}}{q^{2}}=2$
+\end_inset
+
+, siendo 
+\begin_inset Formula $\frac{p}{q}$
+\end_inset
+
+ irreducible.
+ Tenemos que 
+\begin_inset Formula $p^{2}=2q^{2}$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $p^{2}$
+\end_inset
+
+ es par.
+ Por tanto 
+\begin_inset Formula $p$
+\end_inset
+
+ debe ser par porque si fuera 
+\begin_inset Formula $p=2k+1$
+\end_inset
+
+, 
+\begin_inset Formula $p^{2}=4k^{2}+4k+1$
+\end_inset
+
+ sería impar.
+ Sea pues 
+\begin_inset Formula $2p':=p$
+\end_inset
+
+ (con 
+\begin_inset Formula $p\in\mathbb{N}$
+\end_inset
+
+).
+ Entonces 
+\begin_inset Formula $4(p')^{2}=2q^{2}$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $2(p')^{2}=q^{2}$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $\exists q'\in\mathbb{N}:q=2q'$
+\end_inset
+
+, lo que contradice el hecho de que 
+\begin_inset Formula $p/q$
+\end_inset
+
+ sea irreducible.
+\end_layout
+
+\begin_layout Plain Layout
+La siguiente demostración usa la fórmula 
+\begin_inset Formula $(1+\varepsilon)^{n}<1+3^{n}\varepsilon\forall n\in\mathbb{N},0<\varepsilon<1$
+\end_inset
+
+, que demostraremos por inducción.
+ Para 
+\begin_inset Formula $n=1$
+\end_inset
+
+ tenemos que 
+\begin_inset Formula $1+\varepsilon<1+3\varepsilon=1+3\varepsilon$
+\end_inset
+
+.
+ Ahora bien, si se cumple para cierto 
+\begin_inset Formula $n$
+\end_inset
+
+, podemos probar que se cumple para 
+\begin_inset Formula $n+1$
+\end_inset
+
+:
+\begin_inset Formula 
+\[
+1+3^{n+1}\varepsilon=1+3\cdot3^{n}\varepsilon=3\cdot(1+3^{n}\varepsilon)-2>3\cdot(1+\varepsilon)^{n}-2\overset{?}{>}(1+\varepsilon)^{n}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+La última desigualdad se cumple siempre que 
+\begin_inset Formula $2\cdot(1+\varepsilon)^{n}>2$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $(1+\varepsilon)^{n}>1$
+\end_inset
+
+, lo cual es verdad, demostrando la fórmula inicial.
+ Ahora usaremos esta fórmula para demostrar que si 
+\begin_inset Formula $r\in\mathbb{Q}$
+\end_inset
+
+ con 
+\begin_inset Formula $r>0$
+\end_inset
+
+ y 
+\begin_inset Formula $r^{2}<2$
+\end_inset
+
+, existe un 
+\begin_inset Formula $t\in\mathbb{Q}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $r<t$
+\end_inset
+
+ y 
+\begin_inset Formula $r^{2}<t^{2}<2$
+\end_inset
+
+.
+ De igual modo, si 
+\begin_inset Formula $s\in\mathbb{Q}$
+\end_inset
+
+ con 
+\begin_inset Formula $s>0$
+\end_inset
+
+ y 
+\begin_inset Formula $s^{2}>2$
+\end_inset
+
+, existe 
+\begin_inset Formula $w\in\mathbb{Q}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $0<w<s$
+\end_inset
+
+ y 
+\begin_inset Formula $s^{2}>w^{2}>2$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Plain Layout
+Para demostrar la primera afirmación debemos ver que es posible encontrar
+ 
+\begin_inset Formula $0<\varepsilon\in\mathbb{Q}$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $t:=r(1+\varepsilon)$
+\end_inset
+
+ se tenga 
+\begin_inset Formula $t^{2}<2$
+\end_inset
+
+.
+ Pero usando la afirmación anterior tenemos que 
+\begin_inset Formula $t^{2}=r^{2}(1+\varepsilon)^{2}<r^{2}(1+9\varepsilon)$
+\end_inset
+
+, y queda encontrar un 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+ tal que 
+\begin_inset Formula $r^{2}(1+9\varepsilon)<2$
+\end_inset
+
+, lo que se consigue con 
+\begin_inset Formula $0<\varepsilon<\frac{1}{9}\left(\frac{2}{r^{2}}-1\right)$
+\end_inset
+
+.
+ Sabemos que este número existe porque 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+ es un cuerpo denso.
+ La demostración de la segunda afirmación es análoga, pero tomando 
+\begin_inset Formula $w:=\frac{s}{1+\varepsilon}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Plain Layout
+Ahora veremos que esto también se cumple con si 
+\begin_inset Formula $r$
+\end_inset
+
+ y 
+\begin_inset Formula $s$
+\end_inset
+
+ no son necesariamente racionales.
+ Razonamos igual que antes, pero como no sabemos si 
+\begin_inset Formula $r$
+\end_inset
+
+ es racional, tampoco sabemos si lo es 
+\begin_inset Formula $t$
+\end_inset
+
+, pero sabemos que, como 
+\begin_inset Formula $r<t$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\exists\tau\in\mathbb{Q}:r<\tau<t$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $r^{2}<\tau^{2}<t^{2}<2$
+\end_inset
+
+, que es lo que buscamos.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\exists\alpha\in\mathbb{R}\backslash\mathbb{Q}:(\alpha^{2}=2\land\alpha=\sup\{0\leq r\in\mathbb{Q}:r^{2}<2\})$
+\end_inset
+
+.
+ 
+\series bold
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $A=\{0\leq r\in\mathbb{Q}:r^{2}<2\}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $1\in A$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $A\neq\emptyset$
+\end_inset
+
+, y está acotado superiormente por 
+\begin_inset Formula $2$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $\exists\alpha:=\sup A$
+\end_inset
+
+.
+ Ahora debemos demostrar que 
+\begin_inset Formula $\alpha^{2}=2$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\alpha^{2}<2$
+\end_inset
+
+, tendríamos que 
+\begin_inset Formula $\exists t\in\mathbb{Q}:\alpha<t\land\alpha^{2}<t^{2}<2$
+\end_inset
+
+, pero 
+\begin_inset Formula $t\in A\#$
+\end_inset
+
+.
+ Por otro lado, si 
+\begin_inset Formula $\alpha^{2}>2$
+\end_inset
+
+, 
+\begin_inset Formula $\exists s\in\mathbb{Q}:\alpha>s\land s^{2}>2$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $2<s^{2}<\alpha^{2}$
+\end_inset
+
+ y, como 
+\begin_inset Formula $s$
+\end_inset
+
+ es cota superior de 
+\begin_inset Formula $A$
+\end_inset
+
+ con 
+\begin_inset Formula $s<\alpha$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ no es el supremo
+\begin_inset Formula $\#$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $\alpha^{2}=2$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+números irracionales
+\series default
+ a los elementos de 
+\begin_inset Formula $\mathbb{R}\backslash\text{\mathbb{Q}}$
+\end_inset
+
+.
+ Se tiene que si 
+\begin_inset Formula $x,y\in\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $x<y$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\exists z\in\mathbb{R}\backslash\mathbb{Q}:x<z<y$
+\end_inset
+
+.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Sea 
+\begin_inset Formula $w\in\mathbb{Q}:x<w<y$
+\end_inset
+
+.
+ Por la propiedad arquimediana, 
+\begin_inset Formula $\exists n:\frac{\sqrt{2}}{n}<y-w$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $z:=w+\frac{\sqrt{2}}{n}$
+\end_inset
+
+.
+ También podemos probar que 
+\begin_inset Formula $\forall x\in\mathbb{R},x=\sup\{r:r\in\mathbb{Q},r<x\}$
+\end_inset
+
+, pues si 
+\begin_inset Formula $\alpha>x$
+\end_inset
+
+, 
+\begin_inset Formula $x$
+\end_inset
+
+ sería una cota superior menor
+\begin_inset Formula $\#$
+\end_inset
+
+, y si fuera menor, entonces 
+\begin_inset Formula $\exists r\in\mathbb{Q}:\alpha<r<x\#$
+\end_inset
+
+.
+ Por esto a 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ se le llama tradicionalmente 
+\begin_inset Quotes cld
+\end_inset
+
+el continuo
+\begin_inset Quotes crd
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Valor absoluto
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+|x|:=\begin{cases}
+x & \text{si }x\geq0\\
+-x & \text{si }x<0
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|x|=|-x|\geq0$
+\end_inset
+
+; 
+\begin_inset Formula $x\neq0\implies|x|>0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|x|=\max\{x,-x\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|xy|=|x||y|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left|\frac{1}{x}\right|=\frac{1}{|x|}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|x|\leq a\iff-a\leq x\leq a$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+|x|=\max\{x,-x\}\leq a\equiv x\leq a\land-x\leq a\equiv-a\leq x\leq a
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Desigualdad triangular:
+\series default
+ 
+\begin_inset Formula $|x+y|\leq|x|+|y|$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+\begin{array}{c}
+-|x|\leq x\leq|x|\\
+-|y|\leq y\leq|y|
+\end{array}\implies-(|x|+|y|)\leq x+y\leq(|x|+|y|)\overset{(5)}{\implies}|x+y|\leq|x|+|y|
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left||x|-|y|\right|\leq|x-y|$
+\end_inset
+
+.
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+\begin{array}{cc}
+z:=y-x: & |x+z|\leq|x|+|z|\implies|x+y-x|=|y|\leq|x|+|y-x|\implies\\
+ & \implies|y|-|x|\leq|y-x|\\
+z':=x-y: & |y+z'|\leq|y|+|z'|\implies|y+x-y|=|x|\leq|y|+|x-y|\implies\\
+ & \implies|x|-|y|\leq|x-y|
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left|\sum_{k=1}^{n}x_{k}\right|\leq\sum_{k=1}^{n}|x_{k}|$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+Se obtiene por inducción sobre la desigualdad triangular.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Distancia
+\series default
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+ a 
+\begin_inset Formula $y$
+\end_inset
+
+: 
+\begin_inset Formula $d(x,y):=|x-y|$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $d(x,y)=0\iff x=y$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $d(x,y)=d(y,x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $d(x,z)\leq d(x,y)+d(y,z)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Raíces 
+\begin_inset Formula $n$
+\end_inset
+
+-ésimas
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $x\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $x>0$
+\end_inset
+
+ y sea 
+\begin_inset Formula $p\in\mathbb{N}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall r\in\mathbb{Q},r>0,r^{p}<x,\exists t\in\mathbb{Q}:(r<t\land r^{p}<t^{p}<x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall s\in\mathbb{Q},s>0,s^{p}>x,\exists w\in\mathbb{Q}:(0<w<s\land s^{p}>w^{p}>x)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\exists!\alpha\in\mathbb{R},\alpha>0:\alpha^{p}=x$
+\end_inset
+
+; 
+\begin_inset Formula $\alpha=\sup\{r\in\mathbb{Q}:r^{p}<x\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así, la 
+\series bold
+raíz 
+\begin_inset Formula $p$
+\end_inset
+
+-ésima
+\series default
+ de 
+\begin_inset Formula $x$
+\end_inset
+
+ se define como el único número real positivo 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\alpha^{p}=x$
+\end_inset
+
+.
+ Lo escribimos como
+\begin_inset Formula 
+\[
+x^{\frac{1}{p}}:=\sqrt[p]{x}:=\sup\{r:r\in\mathbb{Q},r^{p}<x\}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document