3. Prove the contraction mapping theorem. (Priority) Theorem 0.1 (Contraction Mapping Theorem). Let D be a closed subset of R. If F : D - Dis a contraction, then it has a unique fixed point. Moreover, this fixed point is the limit of the functional iteration starting with any initial guess. 4. Give a counterexample of the contraction mapping theorem if the closedness of D is dropped and explain why your example fails