real analysis - Prove that $limlimits_{ntoinfty} frac{S_n - s}{S_n+s} = 0$ implies $limlimits_{n rightarrow infty} S_n = s$

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.