= 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> $L\geq\sqrt a$. The update is <continuous> on this interval, so passing to the <limit> gives
$$
L=\frac12\left(L+\frac aL\right),\qquad L^2=a.
$$
Positivity 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.
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