To find approximate $\sqrt{a}$ we can use Newton's method to approximately solve the equation ${x}^{2}-a=0$ for $x$, starting from some rational ${x}_{0}$.

Newton's method in general is only locally convergent, so we have to be careful with initialization.

Show that in this case, the method always converges to something if ${x}_{0}\ne 0$.

Newton's method in general is only locally convergent, so we have to be careful with initialization.

Show that in this case, the method always converges to something if ${x}_{0}\ne 0$.