On Jan 3, 7:32 am, George Greene <gree...@email.unc.edu> wrote: > On Dec 29 2012, 5:18 pm, Graham Cooper <grahamcoop...@gmail.com> > wrote: > > > AD METHOD (binary version) > > Choose the number 0.a_1a_2a_3...., where a_i = 1 if the i-th > > number in your list had zero in its i-position, a_i = 0 otherwise. > > > LIST > > R1= < <314><15><926><535><8979><323> ... > > > R2= < <27><18281828><459045><235360> ... > > > R3= < <333><333><333><333><333><333> ... > > > R4= < <888888888888888888888><8><88> ... > > > R5= < <0123456789><0123456789><01234 ... > > > R6= < <1><414><21356><2373095><0488> ... > > > .... > > Well, YES, we KNOW that this is the kind of thing (a square wxw list) > that we are talking about. > > > By breaking each infinite expansion into arbitrary finite length > > segments > > The word "arbitrary" DOES NOT HELP you here. > If it TRULY is "arbitrary" then I can arbitrarily pick ONE as my > segment-length. And then the whole notion of segments JUST GOES AWAY. > It's JUST DIGITS. Your going to segments DOESN'T CHANGE anything. > If there were originally 10 digits (0-9) and you go to length-3 > segments, > then that is JUST LIKE having A THOUSAND digits an STAYING with length- > *1* > segments, i.e., NOT HAVING ANY SEGMENTS AT ALL. > > Dumbass.
*I* provide the List of All Reals.
*I* guarantee all possible segments are listed, in every possible order.
PROOF BY INDUCTION.
BASE STEP A(n) All 10^n segments occur in segment position 1, 2, 3,... infinitely many times each in each position.
INDUCTIVE STEP If <s1> <s2> are 2 consecutive listed segments in BASE STEP < s11 s12 s13... s1m s21 s22 s23 .. s2n>
where m is the length of segment 1 and n is the length of segment 2
is ALSO a segment of the BASE STEP
Therefore by Induction, all sequences of all segments are listed.