Limite de sequência (Unicidade)

$\text{Se} \, \lim_{n \to \infty} x_n = a \, \text{e} \, \lim_{n \to \infty} x_n = b, \, \text{ent\~ao} \, a = b.$

Palavras-chave: limite de sequência, unicidade, definição.

DanielFerreira escreveu: $gif.latex?\text{Se}&space;\,&space;\lim_{n&space;\to&space;\infty}&space;x_n&space;=&space;a&space;\,&space;\text{e}&space;\,&space;\lim_{n&space;\to&space;\infty}&space;x_n&space;=&space;b,&space;\,&space;\text{ent\~ao}&space;\,&space;a&space;=&space;b.$

Inicialmente, supomos, por absurdo, $gif.latex?\mathsf{a&space;\neq&space;b}$ . Assim, consideramos $gif.latex?\mathsf{b&space;>&space;a}$ .

Por hipótese, temos que:

$gif.latex?\mathsf{\forall&space;\epsilon&space;>&space;0}$ , em particular, $gif.latex?\mathsf{\epsilon&space;=&space;\frac{b&space;-&space;a}{2}&space;>&space;0,&space;\exists&space;n_1&space;\in&space;\mathbb{N};&space;|x_n&space;-&space;a|&space;<&space;\frac{b&space;-&space;a}{2},&space;\forall&space;n&space;>&space;n_1}$

e, $gif.latex?\mathsf{\exists&space;n_2&space;\in&space;\mathbb{N};&space;|x_n&space;-&space;b|&space;<&space;\frac{b&space;-&space;a}{2},&space;\forall&space;n&space;>&space;n_2}$ .

Desenvolvendo,

$gif.latex?\\&space;\mathsf{|x_n&space;-&space;a|&space;<&space;\frac{b&space;-&space;a}{2},&space;\&space;\forall&space;n&space;>&space;n_1}&space;\\\\\\&space;\mathsf{\Rightarrow&space;-&space;\frac{b&space;-&space;a}{2}&space;<&space;x_n&space;-&space;a&space;<&space;\frac{b&space;-&space;a}{2}}&space;\\\\\\&space;\mathsf{\Rightarrow&space;-&space;\frac{b&space;-&space;a}{2}&space;+&space;a&space;<&space;x_n&space;-&space;a&space;+&space;a&space;<&space;\frac{b&space;-&space;a}{2}&space;+&space;a}&space;\\\\\\&space;\mathsf{\Rightarrow&space;\frac{3a&space;-&space;b}{2}&space;<&space;x_n&space;<&space;\frac{a&space;+&space;b}{2},&space;\&space;\forall&space;n&space;>&space;n_1}$

Ademais,

$gif.latex?\\&space;\mathsf{|x_n&space;-&space;b|&space;<&space;\frac{b&space;-&space;a}{2},&space;\&space;\forall&space;n&space;>&space;n_2}&space;\\\\\\&space;\mathsf{\Rightarrow&space;-&space;\frac{b&space;-&space;a}{2}&space;<&space;x_n&space;-&space;b&space;<&space;\frac{b&space;-&space;a}{2}}&space;\\\\\\&space;\mathsf{\Rightarrow&space;-&space;\frac{b&space;-&space;a}{2}&space;+&space;b&space;<&space;x_n&space;-&space;b&space;+&space;b&space;<&space;\frac{b&space;-&space;a}{2}&space;+&space;b}&space;\\\\\\&space;\mathsf{\Rightarrow&space;\frac{b&space;+&space;a}{2}&space;<&space;x_n&space;<&space;\frac{3b&space;-&space;a}{2},&space;\&space;\forall&space;n&space;>&space;n_2}$

Tomando $gif.latex?\mathsf{n_0&space;=&space;m\acute{a}x&space;\left&space;\{&space;n_1,&space;n_2&space;\right&space;\}}$ , teremos:

$gif.latex?\\&space;\mathsf{\frac{3a&space;-&space;b}{2}&space;<&space;x_n&space;<&space;\frac{a&space;+&space;b}{2}&space;\quad&space;e&space;\quad&space;\frac{a&space;+&space;b}{2}&space;<&space;x_n&space;<&space;\frac{3b&space;-&space;a}{2},&space;\&space;\forall&space;n&space;>&space;n_0}&space;\\\\\\&space;\mathsf{\Rightarrow&space;\frac{a&space;+&space;b}{2}&space;<&space;x_n&space;<&space;\frac{a&space;+&space;b}{2}&space;\qquad&space;\qquad&space;\qquad&space;\qquad&space;\qquad&space;(Absurdo!)}$

De modo análogo, consideramos o caso $gif.latex?\mathsf{a&space;>&space;b}$ e tomamos $gif.latex?\mathsf{\epsilon&space;=&space;\frac{a&space;-&space;b}{2}&space;>&space;0}$ .

Limite de sequência (Unicidade)

Limite de sequência (Unicidade)

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