This proof seems amazing. I finally understood it after one hour. (I am slow and not mature) I have a question though. In 1.20 (b), The line that says, "Hence there is an integer m (with -m2[tex]\leq[/tex]m[tex]\leq[/tex]m1) such that... How does he justify selecting an m in between the two? I feel uncomfortable with this since it is similar to what is being proved, except for the fact that m is an integer. Is there a Theorem that says we can pick an integer in between two reals?