Prove the proposition. For a successful or failed search in an ordered array with N keys, no more than lg N 1 comparisons are used.

Prove Proposition: Top-down mergesort uses between ½ N lg N and N lg N compares to sort any array of length N.

Question 2: Recurrences (a) Below is the pseudo-code for two algorithms: Practicel (A,s,f) and Practice2(A,s,f), which take as input a sorted array A, indexed from s to f. The algorithms make a call to Bsearch(A,s,f,k) which we saw in class. Determine the worst-case runtime recurrece for each algorithm: T1(n) and T2(n). Show that T(n) is O((log n)²) and T2(n) is O(n log n). Practicel (A,s,f) if s< f ql = L(s+ f)/2] if BSearch(A,s,ql,1) = true Practice2(A,s,f) if s < f if BSearch(A,s+1,f,1) = true return true return true else else return Practice2(A, s, f-1) return Practicel(A, ql+1, f) else else return false return false

Consider this alogrithm:// pre: array A of length n, each A[i] is picked randomly uniformly// from the set {0,1,2,3,4,5,6,7,8,9}.// post: return the index of the first occurrence of "check digit" of A// (already computed in the code)// return -1 if check digit is not foundint FindCheckDigit(A) {check_digit = 0for (i=0; i < n; i++)     check_digit = check_digit + A[i]check_digit = check_digit % 10 # % means remainder of divisionfor (i = 0; i < n; i++) {     if (A[i] == check_digit) {          return i     }}return -1;}Compute the average runtime for this algorithm. Show all details of your computation for both loops.