On Tuesday, 9 July 2013 00:59:26 UTC+2, Virgil wrote:
> Consider the strictly increasing sequence f(n) = (n-1)/n.
> While there is never a last term to that sequence, but when one has increases to 1, one has past by all the terms of that sequence.
The limit 1 will never be reached by passing through infinitely many terms. The limit 1 only says that no term will be 1 but many terms will come as close as you like.
It is impossible to reach pi by its rational approximations. It is impossible to define Cantor's diagonal by digits. All its digits belong to rational approximations. But all rational approximations can be in a rationals-complete list. Therefore Cantor's diagonal argument fails:
For every index n exist infinitely many lines k such that the FIS of the diagonal is in such a line k: d_1, d_2, d_3, ..., d_n = q_k1, q_k2, q_k3, ..., q_kn.
Since it is impossible to find digits of D that are not innfinitely many lines together too, the belief in the existence of d is of the same sort as the belief in the existence of devils.