State and prove the Contraction Mapping Theorem.
Let be a bounded metric space, and let denote the set of all continuous maps . Let be the function
Show that is a metric on , and that is complete if is complete. [You may assume that a uniform limit of continuous functions is continuous.]
Now suppose that is complete. Let be the set of contraction mappings, and let be the function which sends a contraction mapping to its unique fixed point. Show that is continuous. [Hint: fix and consider , where is close to .]