I'm trying to build my intuition regarding the Cauchy-sequence construction of the reals.
Essentially, do you think that it is more accurate to visualize a real number as being defined by sequences of rationals dancing arond it (but never getting to it) as opposed to converging to it (as is conventionally visualized).
The latter image of convergence threatens to 1) introduce circularity to the (albeit intuitive) conception of the Cauchy construction. This is because rational sequences are imagined to converge to a real point but we have no conception of such an object of a "real number" (or a real point). 2) Plus, real numbers are themselves (an equivalent class of) sequences, and it's mind boggling when we think about how a sequence converges to...a bunch of many many (equivalent Cauchy) sequences! (We think of convergence as things narrowing down but here, it's exploding).
So, instead of imagining real numbers as 'the point' to which a Cauchy sequence converge, I wasthinking if we can imagine real numbers as being defined by rational numbers dancing around very close to it (particularly the rational numbers at the tail ends of the real number's equivalent Cauchy sequences). So, it's like a donut. The hole is empty. We don't know what the real number (the hole) is, so we define it in terms of the breading all around that hole.
Would such an imagining be conceptually accurate? Thank you!