The objective of this exercise is to prove that the logarithm is an unbounded function from very basic principles, particularly the geometric intuition behind the integral. You will not get any marks if you identify IT * dt as In x and just mention that In a diverges to infinity as x -+ co. Let L(x) := fi +dt. (a) Fix an integer k > 1. In the interval [k, k + 1], find a lower bound for L. Hint: You may want to find first a lower bound for the function 1/x in [k, k + 1]. (b) Let x, y E R be two positive numbers with x k Fix an integer N > 1. In the interval [1, N], find a lower bound for L, in terms of some harmonic sum. (d) Now you remember that something else happened in your dreams last night. The Oracle of Delphi showed up and told you that limn - H(n) = co, in other words, for each number B > 0 that you told the Oracle, the Oracle was able to mention an integer N > 0 such that for every n > N we have H(n) > B. Use this to show that lim L(a) = 00