On Nov 23, 2:44 pm, Virgil <vir...@ligriv.com> wrote: > In article > <49b3cb68-22ab-423a-ac87-048870090...@w1g2000vbx.googlegroups.com>, > > > > > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > On 23 Nov., 22:19, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > in any case the limit of my sequence > > > > > 01. > > > > > 0.1 > > > > > 010.1 > > > > > 01.01 > > > > > 0101.01 > > > > > 010.101 > > > > > 01010.101 > > > > > 0101.0101 > > > > > ... > > > > has infinitely many digits right to the point as well as left to the > > > point. > > > This can be proved by attaching pairs 01 of digits always from the > > left and from the right side in a symmetrical way: > > > 01. > > 01.01 > > 0101.01 > > 0101.0101 > > ... > > > The overall behaviour of the digits (not of the indices) is the same > > as that of the original sequence. > > The sequence > .01 > .0101 > .010101 > .01010101 > ... > has limit 1/99, which is finitely expressible both as a repeating > decimal and as a rational number, but that does not mean that either > > 01. > 01.01 > 0101.01 > 0101.0101 > ... > > or > > 01. > 0.1 > 010.1 > 01.01 > 0101.01 > 010.101 > 01010.101 > 0101.0101 > ... > > has any limit expressible in the positional notation of decimals, or any > other such base. > --
