Extend example 1.64 to develop an algorithm for approximating the square root of any positive number. The

Question:

Extend example 1.64 to develop an algorithm for approximating the square root of any positive number.
The following exercises establish the links between convergence of sequences and geometry of sets. First, we establish that the boundary of a set corresponds to the limits of sequences of elements in the set. This leads to an alternative characterization of closed sets which is useful in applications.