Re: Relative Cardinality



mueckenh@xxxxxxxxxxxxxxxxx wrote:
> Proginoskes wrote:
> > (You've mixed up quantifiers here. Pi can be defined by a sequence of
> > rational numbers a_n, but the condition on this sequence is that for
> > every positive epsilon, THERE EXISTS AN N such that |a_n - pi| <
> > epsilon when n >= N. The value of N changes when epsilon changes; you
> > do not need to have |a_n - pi| < epsilon for all n and for all epsilon.
>
> You misunderstood me. Perhaps my fault. It must be possible to satisfy
> this inequality for arbitrarily small epsilon.

Yes.

> But already for the
> comparatively large eps = 1/10^10^100 this inequality cannot be
> satisfied by any a_n.

There are infinitely many rational numbers which are closer
than 1/10^10^100 to pi. For instance, all those which are
identical to pi in the first 10^10^100 digits but differ
by 2 in the 1+10^10^100-th digit, satisfy |a-pi| < eps
for your choice of eps.

- Randy

.