Chapter 9.2, Problem 28ES

Let $\left\{a_n\right\}$ be the sequence defined recursively by $a_1=1\text{and}{a}_{n+1}=\frac{1}{2}[a_n+\left(3/a_n\right)]\text{for}n\ge 1.$

(a)Show that $a_n\ge \sqrt{3}\text{for}n\ge 2.$

(b) Show that $\left\{a_n\right\}$ is eventually decreasing.

(c) Show that $\left\{a_n\right\}$ converges and find its limit L.