Using a recursion scheme in which[0,1]is initially sub divided into four equal intervals and the square[0,1]×[0,1]is initially subdivided into four equal subs quares, give an analytic definition for...

Using a recursion scheme in which[0,1]is initially sub divided into four equal intervals and the square[0,1]×[0,1]is initially subdivided into four equal subs quares, give an analytic definition for the function shn:[0,1]→[0,1]×[0,1] in volved in defining the Hilbert curve (see Figure 19.1). Prove that the sequence hn converges to a continuous function h. Prove that the hn can be chosen to be injective but that h cannot be injective.