Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, The Countable Reals) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a topos that makes Dedekind reals countable.

I'm wondering if the constructive Eudoxus reals are sequence-avoiding?

Also, if “all functions $\mathbb{Z} \to \mathbb{Z}$ are computable” holds, what will be the behaviour of Eudoxus reals? Would it become computable?