LudovicoVan wrote: >> But, have you got any proof or argument to support your claim >> that "[it is a] fact that technically speaking the collection >> of ALL real number does exhaust the collection of all infinite >> binary sequences"?
Jack Campin wrote: >> Given any binary sequence f: N ->> {0,1}, define >> >> r(f) = sum (n = 0 ->> inf) 2 ^ (-n) >> >> This is always a real number in [0, 1]. >> >> Done.
LudovicoVan wrote: >> You have done nothing that proves the claim above.
Uirgil wrote: > Do you dispute that every r(f) is necessarily a real number? > That would be the only basis for any objection.
It's true that every r(f) is a real in [0,1].
However the phrase "the collection of real numbers exhausts the collection of all infinite binary sequences" seems to imply that the mapping should be the other way around (although I could simply be reading it wrong); i.e., that every real in [0,1] maps to a unique infinite binary sequence.
Of course, that should not be too hard to show, either. (Except for the possible complication that every rational real has two different binary representations.)
