![]() There are a couple different natural ways you might define a positive general sequence so that Theorem 3 remains true-you could either take the condition of Theorem 1 as the definition, or you could weaken the positive tail condition to say that for any rational $\epsilon>0$, $(a_n \epsilon)$ has a positive tail. ![]() Roughly, you could have a sequence which oscillates between large positive values and positive values approaching $0$, and then a sequence could be co-Cauchy with it while being negative on the part that approaches $0$, so it does not have a positive tail. Note, though, that the characterization of positiveness which you use from Theorem 1 does use the fact that $(a_n)$ is Cauchy-the original definition of positiveness is weaker for general sequences, and Theorem 3 is not true for general sequences with the original definition of positiviteness. It doesn't matter whether your first sequence was actually closing in on any fixed positive number, if all you care about is staying above $0$. Intuitively, it should make sense-if you have a sequence that stays a bounded distance above $0$, then any other sequence getting close to it will also say above $0$. It seems that all these definitions and theorems (except theorem 2) can make sense for arbitrary sequences which are not necessarily Cauchy. Is my argument for the last theorem correct? I am a little skeptical as I didn't use the Cauchyness of $(a_n)$ at all. Indeed, this question arises when one wants to prove the well-definedness of an order while constructing $\mathbb$.
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