4. Prove each existential statement given below. a. There are positive integers x and y such that 1/x + 1/y is an integer. b. There is a positive integer x that is equal to the sum of all the positive integers less than x. c. There are integers m and n such that vm + n = m + vn d. There are three positive integers, x, y, and z, that satisfy x2 + y2 = z2 e. Show that there is an integer n such that n2-1 is prime. f. For every pair of real numbers, x and y, there exists a real number z such that X - Z =Z-y. 1