http://img156.imageshack.us/i/17818455.jpg/ http://img215.imageshack.us/i/53355598.jpg/ http://img509.imageshack.us/i/11493310.jpg/ If you look at the above, I have underlined the problem that I am having. So, my first question is, where are these inequalities coming from? If you do have other questions involving such approach, please show me. My other question is from the explanation of Riemann sum, I do not understand the sign "llPll " thing and I am having trouble understanding Riemann sum and how it really is related to the upper and lower sums. Thank you for your attention.