Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$

Hint You can simplify a) by applying AM-GM, obviously showing that each $S_n>0$ which is easy to show $$\frac{1}{2}\left(S_n+\frac{A}{S_n}\right)\geq \sqrt{S_n \cdot \frac{A}{S_n}}=\sqrt{A}$$ For c) more details here and you already proved the sequence is decreasing and bounded below in b) and a).

For d) as Eugen suggested, with a), b) and c) you are confident that the sequence is converging so: $$\lim_{n \rightarrow \infty} S_{n+1}= \lim_{n \rightarrow \infty} \frac{1}{2}\left(S_n+\frac{A}{S_n}\right) \Rightarrow s=\frac{1}{2}\left(\lim_{n \rightarrow \infty}S_n+\frac{A}{\lim\limits_{n \rightarrow \infty} S_n}\right)\Rightarrow \\ s=\frac{1}{2}\left(s+\frac{A}{s}\right)$$ obviously $s \geq \sqrt{A}>0$, thus $$2s^2=s^2+A \Rightarrow s=\sqrt{A}$$