Question

Suppose lim n→∞ (sn −1)/(sn +1)=0. Prove that limn→∞sn = 1

Answer 1

Here we have,

$$\lim\limits_{x\to\infin}\frac{s_n-1}{s_n+1}=0\\~\\ \Rightarrow \lim\limits_{x\to\infin} \frac{s_n+1-2}{s_n+1}=0\\~\\ \Rightarrow \lim\limits_{x\to\infin}(1-\frac{2}{s_n+1})=0\\~\\ \Rightarrow 1-\lim\limits_{x\to\infin}\frac{2}{s_n+1}=0\\~\\ \Rightarrow \lim\limits_{x\to\infin}\frac{1}{s_n+1}=\frac{1}{2}$$

Now, as this is an equality and RHS is finite, we can be sure that $\lim\limits_{x\to\infin}s_n$ is some value let say $c\in\R$ .

i.e. $c=\lim\limits_{x\to\infin}s_n$

Therefore,

$$\frac{1}{c+1}=\frac{1}{2}\\~\\ \Rightarrow c+1=2\\~\\ \Rightarrow c=1$$

So, we have:-

$$\fcolorbox{black}{aqua}{$\textcolor{black}{\lim\limits_{x\to\infin}s_n=1}$}$$