Neighbor says: >I mean to say that when we want to divide AB in half, we should divide it in AM and NB, with M and N two consecutive points
OK, for a while I thought maybe you were putting us on (what with your fake moniker and all) but I'm guessing now you are sincere. "Two consecutive points" has no meaning in the real number based continuum, or even in the set of rational numbers with usual ordering. So, if you cannot accept that between two real (or rational) numbers there is always another (which means that "two consecutive" simply does not exist in these games) then you will never accept the more or less accepted answers to your questions.
So, lets simplify a bit: can you accept that between two points their is always another? Why or why not? If you do accept that, then what possible meaning does "two consecutive points" have? Any two you pick, trying to get that elusive consecutive pair, will always have another between them. Therefore they are not consecutive.