As part of this quora post(http://www.quora.com/What-is-Knuths-optimization-in-dynamic-programming) a description of knuth optimization has been given. It has been mentioned that :- it can be proven that mid[l,r-1] <= mid[l,r] <= mid[l+1,r] — this means monotonicity of mid by l and r. Can anyone of you please help me regarding how to prove it?