Don't you need to quotient after you've got Cauchy sequences? For example, (0,0,0,0,....) is a representative of the same real number (as Cauchy sequence) same as (1,0,0,0,0,0,....), but both are different as streams.
Yeah, depends on how you're encoding it. Quotient types are to my knowledge still tricky to implement, but you can try to make a setoid and structure this type with some kind of equality. Of course, now we run directly into totality and semi-decidability.