 Solution

After the first update, the sequence lies in $[\sqrt{a},x_1]$ and is nonincreasing. The monotone convergence theorem for sequences gives a limit $L\ge \sqrt{a}$. The update is continuous on this interval, so passing to the limit givesPositivity forces $\boxed{x_n\to \sqrt{a}\text{ for every }{x}_0>0}$. If an iterate reaches the root exactly, the constant tail gives the same conclusion. The Exact Newton error identity for a quadratic here is $x_n-\sqrt{a}=(x_{n-1}-\sqrt{a}{)}^2/(2x_{n-1})$, also showing quadratic convergence once the error is small.