Help with highlighted part iv.

Consider the equation 1+u2 0 { 3&0) i. Consider the map (I), which maps f to the following function of :C: [I 1 + f(t)2dt. 0 Prove that if f is in the set X : {f E (10%] : m 3 1}. then (EU) is also in X. ii. Prove that X as above is a complete metric space. iii. Prove that if f1, f2 are two functions in the set X , then d(q)(fl)= q'ffl) S d