I was reading section 2.8 on Newton's method of Hubbard's Vector Calculus,Linear Algebra, and Differential Forms. At the end of this section, it is stated that:
If the sequence $a_0,a_1,a_2...$ converges to a root, the number obtained by squaring the second partials (evaluated at the point $a_0$ ( and then at $ a_1,a_2...)),$ adding them together,and taking the square root of the sum ,steadily decreases and tends to $ 0$.
Why does this sequence of numbers steadily decrease and tend to $0$?