Is there a positive real number which can be written as a Cauchy sequence, such that this Cauchy sequence is bounded away from zero and also this sequence contains infinite number of positive and negative rationals.
e.g. zero can be written as a Cauchy sequence of,
$0.1, \ -0.01, \ 0.001, \ -0.0001, .... $
which has infinite number of negative and positive rationals.
Is this possible for a positive real number ?
I think it is not possible since the difference of two term, $|a_m - a_n|$, in a sequence will never tend to zero since the limit is a positive number ?
Is there a proof which does not use terms 'positive number' and 'limit' ?