Is it possible that a positive real number is a limit of some Cauchy Sequence of rational numbers containing both positive and negative numbers?

Question

The book by Terence Tao defines positive real numbers as limit of Cauchy sequences containing only positive rational numbers. But suppose we have following series:

-1,1,1.1,1.11,1.111,....

Will the limit of this series be a positive real number?

Answer 1

Yes. The limit of that sequence is $\frac{10}{9}$. But it doesn't witness that $\frac{10}{9}$ is a real number according to the definition you have given.

Answer 2

Yes, the limit of your sequence is 10/9 which is a positive number.If you drop the first term of your sequence then you will get a sequence of rationals with all positive terms approaching 10/9. Notice that the presentation of a positive real number as the limit of a sequence of rational numbers is not unique therefore you may have sequences with finitely many negative terms having the same positive limit. You may drop all the negative terms and the limit will not change.The if and only if statement in Tao's definition does not cause any conflict.