The problem involves proving that the rational numbers, as a subset of the real numbers, can be contained within open intervals whose total width is less than any given positive epsilon. The discussion revolves around concepts of countability and convergence in the context of real analysis.
The discussion includes attempts to formulate a series that converges to epsilon, with participants sharing potential series expressions. There is an ongoing exploration of different approaches to represent the total width of the intervals.
Participants are navigating the constraints of the problem, particularly the requirement that the total width of the intervals must be less than any epsilon greater than zero. The implications of countability and convergence are central to the discussion.
zhang128 said:ups, my bad. I was thinking that since rationals are countable, I just need to make the open interval around each rational such that the total sum is less than \epsilon. Then the question turns to prove the sum of a finite series converges to the epsilon??
zhang128 said:hmmm, like epsilon/(n^2+n)??