I was trying to do some homework problems and one of the questions i didn't get was this: Show that x-1/x </= ln x </= x-1 for all x>0. Using mean value thm. Now in class teacher gave us a lemma that would help solve this problem, which is: Suppose that f(a)=g(a), and that f & g are continuous on [a, infinity). Suppose also that f'(x)<g'(x) for all x in (a, infinity). Then f(x) < g(x) for all x in (a, infinity). The only thing i understand for now is that, by graphing the two functions, there seems to be a relation between the gaps of the two functions crossing at a single point, say a. So on the right side of the cross section, one function increases faster while other decreases, or so, etc. but how would i prove this using the lemma, i dont even have proof of the lemma.