Prove that if $$ \lim\limits_{n \rightarrow \infty} \frac{S_n-s}{S_n+s} = 0$$ then $$\lim\limits_{n \rightarrow \infty} S_n = s$$
Hint: Define $t_n = \frac{S_n -s}{S_n + s}$ and solve for $S_n$
By the hint: $$t_n = \frac{S_n -s}{S_n + s}$$ $$(S_n + s)t_n = S_n -s$$ $$S_n(t_n-1)= - s -st_n$$ $$S_n= -s \cdot \frac{1+t_n}{t_n-1}$$
$$\lim\limits_{n \rightarrow \infty} S_n= -s \cdot \frac{1+\lim\limits_{n \rightarrow \infty} t_n}{\lim\limits_{n \rightarrow \infty} t_n-1}$$
As $ \lim\limits_{n \rightarrow \infty} \frac{S_n-s}{S_n+s} = \lim\limits_{n \rightarrow \infty} t_n = 0$, it follows:
$$\lim\limits_{n \rightarrow \infty} S_n= s$$
Is my argumentation correct/appropriate?Anything needs to be added?
Much appreciated for your input.