> > The limit of > > 1 > > 23 > > 456 > > ... > > could be different from > > > 1 > > 12 > > 123 > > ... > > > and what about > > > 1/1 > > 2/1, 2/2 > > 3/1, 3/2, 3/3 > > ... > > > Could the limit of numerators and denominators be different?
> A sequence's limiting behavior is governed not only by the terms of the > sequence, but also by the environment in which that sequence exists. > > A sequence of rationals may not converge in the set of rationals but > still converge in the set of reals. > > So that without a specified environment, no sequence can be claimed to > 'converge' to anything.
Here is no convergence of the magnitudes of numbers under discussion but the convergence of cardinal numbers. No considertion of environment is required other than the foundations of set theory.
