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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' ?

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    Note that the definition of positive real number you are given involves the notion of a limit, so anything you prove is likely to have to refer to the notion of a limit somewhere.2012-11-27

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