It is equivalent to Dedekind cuts as one of my papers shows. You can think of Dedekind cuts as collecting all the lower bounds of the intervals and throwing away the upper bounds. But if you think about flushing out a Dedekind cut to be useful, it is about pairing with an upper bound. For example, if I say that 1 and 1.1 and 1.2 are in the Dedekind cut, then I know the real number is above 1.2. But it could be any number above 1.2. What I also need to know is, say, that 1.5 is not in the cut. Then the real number is between 1.2 and 1.5. But this is really just a slightly roundabout way of talking about an interval that contains the real number.

Similarly with decimals and Cauchy sequences, what is lurking around to make those useful is an interval. If I tell you the sequence consists of a trillion approximations to pi, to within 10^-20 precision, but I do not tell you anything about the tail of the sequence, then one has no information. The next term could easily be -10000. It is having that criterion about all the rest of the terms being within epsilon that matters and that, fundamentally, is an interval notion.