Certainly in Agda I've personally proved that the "Cauchy sequences of rationals" formulation is a partially ordered ring.
However, I believe (but have not tried it and have not proved) that if you stick with Agda's incredibly strict and purely computational logic, the proof does indeed fall over. The problem is that Bob can't identify in general whether one real is less than another (indeed, this would imply solving the halting problem), so he can't know whether he's allowed to pick s_n at time n or not: he can't know whether s_n is less than his current number and hence a valid choice.
However, I believe (but have not tried it and have not proved) that if you stick with Agda's incredibly strict and purely computational logic, the proof does indeed fall over. The problem is that Bob can't identify in general whether one real is less than another (indeed, this would imply solving the halting problem), so he can't know whether he's allowed to pick s_n at time n or not: he can't know whether s_n is less than his current number and hence a valid choice.