In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 16 Jun., 11:48, "|-|ercules" <radgray...@yahoo.com> wrote: > > "WM" <mueck...@rz.fh-augsburg.de> wrote ... > > > > > > > > > By induction we prove: There is no initial segment of the (ANTI)diagonal > > > that is not as a line in the list. > > > > Right, therefore the anti-diagonal does not contain any pattern of digits > > that are not computable. > > > Sorry, you misquoted me. I wrote: > By induction we prove: There is no initial segment of the diagonal > that is not as a line in the list. And there is no part of the > diagonal that is not in one single line of the list. > But I have to excuse because I wrote somewhat unclear. > > The meaning is: > 1) Every initial segment of the decimal expansion of pi is in at least > one line of your list > 3. > 3.1 > 3.14 > 3.141 > ... > What we can finde in the diagonal (not the anti-diagonal), namely > 3.141 and so on, exactly that can be found in one line. This is > obvious by construction of the list. > > 2) Every part of the diagonal is in at least one line. That means, > every part is in one single line, or there are parts that are in > different lines but not in one and the same. > > The latter proposition can be excluded.
How is the latter excluded?
Consider the sequence f(n) = trunc(pi*10^n)/10^n
While there are some successive lines which are equal (where the decimal expansion of pi has 0 digits), for every n there is an m, with n < m, such that f(n) < f(m) < pi.
> If there are more than one > lines that contain parts of pi, then it can be proved, be induction, > that two of them contain the same as one of them. This can be extended > to three lines and four lines and so on for every n lines.
False for my example above. > > Hence we prove that all of pi, that is contained in at least one of > the finite lines of your list, is contained in one single line.
False for my example above.
And equally false for f(n) = trunc(x*10^n)/10^n with any real x which has a non-terminating decimal expansion.