There's two related arguments you (may) be making. One is philosophical, and to that I say: can you give an algorithm to compute a0, a1, ...? If you can't (and I suspect you can't), I'd be hesitant to say that such a sequence exists. But I don't think there's a useful argument to have here ;-).
But more concretely: no, the computable numbers aren't complete. Why would this make analysis messy? I didn't find the presentation and results in Computable Analysis (the book) all that different from what you find in an ordinary analysis course. Is there something it makes hard which I'm missing? (This might be impossible to answer without specific knowledge of Computable Analysis; remember that the definitions of, e.g. limits, are different.)
But more concretely: no, the computable numbers aren't complete. Why would this make analysis messy? I didn't find the presentation and results in Computable Analysis (the book) all that different from what you find in an ordinary analysis course. Is there something it makes hard which I'm missing? (This might be impossible to answer without specific knowledge of Computable Analysis; remember that the definitions of, e.g. limits, are different.)