Question

$[1,\infty)\subseteq A\subseteq\mathbb{R}, \ f\in\mathbb{R}^A$ ve $L\in\mathbb{R}$  olsun. $$\lim\limits_{x\to\infty}f(x)=L\Rightarrow \lim\limits_{n\to\infty} f(n)=L$$ olduğunu gösteriniz.

Comments

$f \in \mathbb{R}^{A}$ ne demek ? $f$ $\mathbb{R}$ kumesinden $A$ ya mi gidiyor demek ?

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$$Y^X:=\{f|f:X\to Y  \text{ fonksiyon}\}$$

Answer 1

$\lim\limits_{x\to\infty}f(x)=L$  ve  $\epsilon>0$  olsun.

$\left.\begin{array}{rr} \lim\limits_{x\to\infty}f(x)=L \\ \\ \epsilon>0 \end{array} \right\}\Rightarrow \begin{array}{c} \\ \\ \left. \begin{array}{rr} (\exists M>0)(\forall x \in A)(x>M\Rightarrow |f(x)-L|<\epsilon) \\ \\ N:=\lfloor M\rfloor +1 \end{array} \right\} \Rightarrow \end{array}$

 

$\Rightarrow (N\in\mathbb{N})(n\geq N\Rightarrow |f(n)-L|<\epsilon).$