I think their point is that if you have a convergent sequence of fractions, you can drop some finite number of them from the start and still converge to the same real number. In some sense, the rest of the sequence already contains all of the information, so early terms are redundant.

With the decimal and continued fraction representations, that's not true. The later parts of the representations only describe refinements on the earlier parts; they don't themselves contain the earlier parts.